How do you minimize dealing with winners that are a pain?

Higher levels. Simple as that. And group only for better games. I wouldn't mind giving to more people who can't give games themselves, but it has turned into an abuse of system by the most greedy. So I just don't bother. Frankly getting past Level 2-3 is really simple and does not require a lot of resources. I did that while I was a student and on a tighter budget.

They have a whole week to do that, sure, but what if the key expires within 24 hours and it's stated as such in the GA description?

I say this as a neutral advice: don't make a giveaway for it. Gift it to a friend, throw it into the orphan keys, make a GA to "delete if key not working" group or something. 5CV for a $25 bundled game doesn't worth getting angry about. If you're talking about your latest giveaway, it's 0CV. If you don't get absolutely nothing from it, it's not setting up a potential source of stress and problems. I hope the current situation gets resolved, and you'll be able to avoid it happening in the future

Don't make giveaways for keys expiring in 24h hours. Or if Steam sale is ending... Simply not worth it.

Also, I don't do public giveaways, just post them on forums and there is surprisingly few issues.

There are many people who have SG open in a background tab in their browser, so it shows online. If there are too many tabs, people forget or don't pay attention.

I have a mixed feeling about your post. Sure, it is sometimes annoying to wait for winner to activate their win, and people who seems active on site but don't activate it makes you doubt if it's just a bot, and seeing people only winning but not giving may feel bad sometimes... But at the same time it looks like you are just over-reacting. There are rules of the site, they are far from perfect, but that's all we have. And rules say that winner have the week to activate a key, so you should comply with that - like, if you see that key may expire or discount in steam may end before the end of week - don't do a giveaway, or make it for some closed group you're sure will be there to activate it in time. Don't blame someone just for following the rules. And once again, I understand, that not activating without explanation and not responding to messages looks impolite and fishy, but at the same time I was in a situation when I didn't have good internet and could not activate my win in a timely basis, and since internet was bad, but not absent - I was actually trying to do this (so probably I appeared as "online" on site), but was unable to either activate key or respond. That was very anxious situation.
As for the worries of getting "not received"... Well, apart from the fact that not activated key is nothing bad, there are already rules for this situation too. If winner didn't view the key during a week - you can just reroll. If they viewed a key and still marked as "not received", or not marked at all - you can file a ticket to request positive feedback, and they will get punishment. Of course, there are still loopholes for all that, as I said - rules are not perfect, but luckily those cases are rare.

For those that miss it at the top, OP basically said he's worried what if his key expires in 24 hours(for whatever reason that would ever happen, I don't even know where you could buy keys that expire in 24 hours), and that this is a whitelisted individual, who last gave away 2 years ago and his last 2 gifts were never received by the winners.

If you think the key can expire in 24 hours, maybe it's not the right key to give away as the rules give people 7 days to activate them, and even then you'd have to try and contact the user, take a screenshot, make a ticket, wait for the mods to look at the ticket, etc. I don't know if it's reasonable to expect people to activate keys in 1 day because some people might have sg to open with the browser which would show them as logging in every day, but they might just close the tab immediately or something and might not actually check it often. I've seen that happen with a few of my winners and honestly there's been some I gave like 2 months to activate their key, and they eventually did or I had to make a ticket.

Also not sure if that's a new whitelisted person or they've been on it for a while, but if it's a recent one, I'm wondering if you considered the red flags.

So from my side, I don't really have this issue, because I don't consider it the end of the world. They have 1 week, I'll give them a week or if I forget, 2 months or whatever lmao. Only after I consider them to be completely absent I request a new winner. Only once I remade a giveaway after like a week and the key that I had to redo ended up as "duplicated" so naturally the previous winner was BL'd.

Occasionally you will have a bad experience, that can't be prevented. Just try to find a balance that you are comfortable with.
Don't use games that are dear or costly to you for a low level public giveaway. If you want some kind of reassurance that someone will (activate and) play your game, use groups such as Playing Appreciated.

And generally speaking, just try not to bother too much. If you gift with no expectations whatsoever you'll enjoy the ride much more.

P.S.
You won't get any CV for 12 is Better Than 6 anyway, so it doesn't really matter in this specific case if they'll activate it in time. And a red mark is irrelevant too.

We can't eliminate those that are constantly on, off, on and yet ignore the fact that they have a win (or maybe two to pick up). Those are the ones that bother us, but they have 7 days to respond, so you are doing all the right things.
I recently had a winner who had not given any gifts away, so I could not contact him on Steamgifts. His comments section was private, so I was unable to contact him via steam. Ironically the comments visible were all from previous gifters who were asking him to pick up his wins :(
After contacting support I was told to email him ... at which point the 7 day wait starts again.

So my black list grows, lol.

Most of my giveaways are public. Like a few have mentioned, there are a couple of groups that specifically aim to match wins to those that will play the game and this is monitored. Also mentioned is the prevalence of the above behaviour (ignoring a request to pick up a win, having to be chased to get a response) tending to be mainly from low levels.
If this behaviour really gets to you then perhaps, send an email on day 1, maybe leaved a message, and if they don't respond then just send a reroll request to support. Support need a screenshot of the email, and links to your prompt to pickup the win ... support will do the rest.

This site only works because gifters like you gift the games. There are some incredibly generous gifters out there, and some lovely winners too, most thank you (and I never check to see if the game has been played, am guilty of an absolutely tragic backlog myself). The few that grind your gears are indeed few, just try to move on and forget them

Here's how I see it.

They have a whole week to do that, sure, but what if the key expires within 24 hours and it's stated as such in the GA description?

SG rules outweigh your GA description. Plus some folks just don't read the descriptions.
They have a week to redeem the key. If the key expires before a week has passed, then unfortunately you set yourself up for this problem.
However. If they take more than a week to provide feedback on the GA, you can contact support. Depending on the situation, they can forcefully mark the GA as Received. (though that should be a last-ditch effort after you've already attempted reasonable forms of communication with the winner)
Keys with a short expiration are risky as GAs and are better off being handed out as a freebie in a thread.

their latest public GA, which is from 2 years ago, believe it or not
their two latest GAs are from 2 full years ago and both are marked as Not Received, whereas their wins are as recent as this week

I'm gonna be real with you. How recently they've created a GA is irrelevant to the problem at hand.
What is relevant is those recent wins. Do they have a history of not redeeming their wins? If their win history is mostly green, then there's a good chance they'll get to you Soon™️.

I don't want a Not Received mark

This is the part where I think I can try to help you out the most. Take a deep breath. I know that sucky feeling of having a mark on your record. What's important to remember is that it's actually not a big deal. You're not going to be banned from the site for making one (1) minor mistake. In fact, I would wager you won't see any punishment at all. IF a red mark even happens, just take it as a learning experience. Everyone makes mistakes and that's ok! You've learned from it. And you've resolved yourself to avoid repeating the same mistake. There's no shame in that.

I'm a bit confused if I should answer a question or if that would be considered brerating...
Anyway, I'd never create a giveaway for a game/key that expires within the 7 days. Seems like an obvious fix for the 'problem'.

Totally not a fan of hosting public GA, one can never be sure of any tools or scripts used by the entrant. There is also the issue of getting winners with leachy stats( 500 wins 10 gift sent). I would hypothesize that folks with leachy stats are more casual with SG and therefore more likely to delay confirmation.
Id suggest going with SGtools enabled invite only GAs if winner confirmations is ur main issue, my personal experience has been phenomenal. Not only do I get prompt confirmations from winners but also credible entrants. Best of both worlds.💚