How many client/services exist for PC?

PC is the platform.

Blizzard App, GOG Galaxy, and the Epic Games Launcher just to name a few more.

Not sure if "Platform" is the word your looking for, But I get what your saying.

battle.net
bethesda
arena.net
GOG

and of course... torrents xD

Battle.net
Epic Games
Origin
Uplay
GOG
Arena.net
ARC

And some other smaller ones.

Too many.

GOG
Uplay
Origin
Battle.net
Microsoft
Bethesda
Discord
Twitch
Epic Games
Rockstar
Arena.net
itch.io

And while it doesn't have a launcher, I think Humble should also qualify.

I'm probably even forgetting something.

I think what you are looking for is game clients. If you do a search for PC game clients, you will get a bunch of results for websites that rank the best or most popular Steam alternatives.

I take it that you mean stores where you can buy & download games (that don't just offer keys for other services)?

Apart from the ones mentioned earlier we also have places like:
Gamer's Gate
Big Fish
Humble Bundle (although it's mostly a key shop, they do have some games you can download from there)
itch.io

Windows, Mac OS, and Linux as far as current platforms.

If you mean available game launching clients, then:

• Steam (Valve)
• Battle.net (Activision—Blizzard)
• Origin (EA)
• UPlay (UbiSoft)
• itch (itch.io)
• Bethesda Launcher (Zenimax)
• Epic launcher (Epic)
• GOG Galaxy (CD Project Red)
• Twitch (amazon)
• Microsoft Store (Microsoft)
• Discord app store (Discord)
• Playism (Active)
• Big Fish Games App (BFG)
• Beamdog Client (Beamdog)
• Paradox Launcher (Paradox Interactive)
• Social Club Application (Rockstar/Take Two)

Some of these and a few others sell games with custom or DRM-free installers, or ZIP files (GameHouse, Humble Bundle, WinGameStore, GamesPlanet, GamersGate, Amazon Digital Store, and so on), but they are not exactly desktop clients.

If you mean digital stores for PC games, then too many.

Good thing that there is a great app to keep track of your games over multiple plattforms: playnite

You can add:

• Arc; Neverwinter
• Trion Worlds/Glyph (owned by Gamigo); Defiance, Trove
• Game Session; many games already on Steam
• Wargaming.net; World of Tank (& others)
• Gaijin; War Thunder

And many more exists, lots for F2P games.

facebook gameroom , garena , gamejolt desktop app , itchio desktop app and kongregate kartridge...
minecraft and world of tanks use their own standalone client/launcher...
subscription-based service like utomik , playonjump and oneplay...

riot games(owned by tencent) of course

Heh, this made me look for Get Games. Apparently closed a couple of years back. Good thing they were nice enough to replace my Borderlands download with a Steam key. Glad I managed to get that before they closed.

Yeah I think that's it. So about 30 clients.

Desura

oh wait

tencent WeGame is coming to global pc gaming market too

https://alternativeto.net/software/steam/

And they all want to launch when Windows starts. Hope you like long boot times.