The moral problem lies in justice,

should be

The rightness problem lies in fairness

It certainly doesn't sound as clean, but it's more correct; Viruses are amoral, and unless the pandemic was targetted, no outside morality [objective/persistent] or ethics [subjective/relative, often adds circumstantial exceptions to moral standards] elements are involved. Rightness is the state of being right [in a philosophical sense, "of the good"], and is the underlying principle of morality, ethics, and righteousness [morality as it relates to adhering to a code of conduct such as laws or religious precepts].

Typically you'd skip the rightness reference altogether, and simply state "Of course, with the nature of the pandemic, there's a significant problem in the fairness of its impact." If you do feel inclined to add a modifier (again, not needed) I'd recommend "ethical problem", as you could still relate ethics [in the sense that "perhaps countries should do more to balance out the socioeconomic factors presenting disparities in the impact of the virus between countries"]. However, that'd shift the meaning of your statement from "it's unfair in a cosmic sense" to "it's unfair how countries don't do more aid".

Justice is potentially even more out of place: Justice, at heart, relates to reactions that are deserving relative to the instigating actions- thus not really fitting with the concept of a pandemic; Again, not unless it was targetted, or one is relating it to the downfall of a corrupt government due to their mishandling of corona, or similar. I do say "potentially", however, as you could be referring to a "divine justice" or "karmic justice".

However, as is necessitated by the use of "justice", both are still concepts related to reactions pursuant to actions. It'd be a convoluted or very tangential consideration to relate <suitable punishment or reward for actions> to <poorness>, especially as the presented phrasing (as it relates to English syntax) suggests* the justice is being inflicted upon the poorer countries. Fairness, on the other hand, is directly based in the comparison of contrasting considerations. Ergo, it'd be the appropraite term.

* It's easier to understand by reformatting the structure of the sentence, as follows: "The problem lies in the justice of it: The poorer countries always deal worse with the pandemic." Properly, to convey your sentiment, the formatting should instead be: "The problem lies in the injustice of it: The poor countries always deal worse with the pandemic." As the negation isn't present, the sentiment is reversed.

I wish wealthier countries could send the 3rd dosage to poorer countries to at least balance a bit.

As per previous studies, vaccines take two weeks after a dosage to create a full immune response, which then takes an extra week to build up fully. This is the basis for the three week minimum delay between vaccination doses- though it's worth noting studies also found that delaying for up to two months would puportedly allow the immune response to continue developing (thus indicating that "fully" isn't "fully", just a designated point where rapid development ceases and continued gains are relatively marginal).

As the priority is to maximize vaccination (two doses bring vaccination response from like 65% to like 90% [some estimates for some of the vaccines go as high as 94%], creating an urgency to have both doses) the minimum standard is encouraged. Comparatively, delaying the second dose apparently would only garner you an additional [estimated] 3-4% resistance following the second dose. It's great if you can hold out for it, but not typically worth risking a quicker 35-40% for. Due to that focus on rushing to get the second vaccinations- and also due to the fact that the immunoresponse for any number of vaccination doses will start to deteriorate a bit after around 4 - 6 months, a booster is encouraged at that time. This booster will not only bring the declining immunoresponse back up to peak, but it can also make up for the lack of delay between the initial shots by increasing the overall response similarly. Finally, it can be reinforced to deal with variant strains.

It is however worth noting that, unless you're someone with a weakened immune system, you're still looking at maybe 87% resistance even after several months. This is why the booster is encouraged only for immunocompromised individuals, at present. Meanwhile, in poorer countries, the issue is one of basic vaccination/herd resistance, not of a need to add additional resistance. Ergo, it'd be much more valuable to encourage and support initial shots, rather than booster shots. To that end, it's worth noting that all of the providers have been sending vaccinations to third world countries- though, it's also worth noting, that their response ranges from free donation, to no-profit, to reduced for-profit**. Of course, aid groups and governments are paying any costs, not individuals, but- depending on the pharma company- it definitely does emphasize the disaparity of fairness between nations. Of course, it's a pretty big deal that it's being provided at all, when put to historical considerations.

** I can't find the sources I previously looked at but, IF I recall correctly: Pfizer = For Profit, Moderna and J&J = At-Cost or Reduced Profit, AstraZeneca = Donated some on their own dime. Again, this is based on somewhat hazy recollections at this point, but I believe it's accurate. Note that ALL the brands listed have had donations sent to outside countries, thanks to EU donations of leftover vaccines or the US deliberately making purchases for the sake of donations. The only distinction is how much the brands charged the governments, and if the brands offered any relief on their own dime.

but I quite liked the pandemic. Without wars we need another "violent" thing to terminate some of us.

I'm not going to address your philosophical viewpoints or ideologies themselves, but "appreciated" may be a better term than "liked", assuming your premise is based in utalitarianism benefits, and not in the satisfaction of human demise. The distinction between word usage isn't typically strict enough to make it of special note on its own [thanks to the first usage of "Like", below], but given the context you're presenting, it's probably best to present your perspective as respectfully and clearly as possible [given that Like more typically associates to the latter two of its below usages].. unless, of course, the utalitarian perspective isn't matched to your intent to begin with.

Like: To find agreeable, enjoyable, or satisfying
Appreciate: To recognize or understand [the value of], to recognize with gratitude, to value or admire