No, a monotheistic god is not necessarily omnipotent.
What properties must something have if it is to be an appropriate object of worship, and if it is to provide reason for thinking that there is a reasonable chance that the fundamental human desires just mentioned will be fulfilled? A natural answer is that God must be a person who, at the very least, is very powerful, very knowledgeable, and morally very good.
(And from there "very" becomes "omni" with no justification.)
Maybe this is a bit harsh, but this is nonsense.
Appropriate object of worship
All that's required for a god to be worshipped is for people to (a) believe they may receive some significant benefit in exchange for worship or (b) be grateful for some benefit they believe they already received (e.g. having been created). Frankly speaking, people may worship simply for the benefit of maintaining a status quo or because worship feels good, although these forms of worship may be less appropriate or justified (side note: this form of belief and worship may e.g. have originated from an earlier time, when events could be attributed to a god, yet we later found natural explanations for them).
The author of the above sentences may feel a significant benefit or gratefulness isn't a sufficient criteria for worship, and they're entitled to that opinion. But that doesn't mean no-one will worship given those criteria. I would probably also challenge anyone claiming they will only worship an all-powerful all-knowing and perfectly moral god, because this would imply giving up eternal life if that's offered by a good who doesn't have all 3 of those traits.
I'm reminded of the first episode of The Outer Limits (1995), where a scientist sets up a habitat for alien creatures, who worship him at some point. The scientist is certainly not all-powerful, all-knowing or perfectly moral, but the creatures nonetheless worship him. This fictional example certainly doesn't prove anything, but it does provide an example of worship that's presumably out of gratefulness and/or for some potential future benefits (for him providing the habitat, feeding them and protecting them, and continuing to do so), and the reason for worship should be fairly easy to understand and empathise with.
There are arguably cases of people worshipping other people (e.g. some dictators and cult leaders).
One can also note that it's possible for a god to exist that isn't "an appropriate object of worship". Believing that a god exists is distinct from whether you worship said god, and the former would meet the definition of monotheism.
Human desires
The quote also mentions the chance of a god to fulfilling the "fundamental human desires" mentioned earlier: "that good will triumph, that justice be done, and that the world not be one where death marks the end of the individual’s existence".
This does not, however, seem to have much to do with the question of whether a god exists or which traits they have. We may desire that justice be done with all our heart, but that doesn't mean a hypothetical god can or will fulfil that desire.
This just stinks of the appealing to consequences fallacy, i.e. believing that a god exists because that would imply that justice will be done, which would be like believing I'm rich because that would imply that I can buy a mansion.
Traits of a creator
So worship has a much lower bar than all-powerful, all-knowing and perfectly moral, and desires are irrelevant. But we might also ask whether a hypothetical god would only be capable of creating the universe, and would've done so in the way they did, if they were all-powerful, all-knowing and perfectly moral.
This seems to fall at the first hurdle. There is a lot of evil in the world, which is exactly what gives rise to the problem of evil that you linked to. If a god lacked one or more of those traits, there is no such problem, so this would be much more compatible with reality as it is.
More concretely, I see no reason a perfectly moral being would be needed to create an imperfect world.
As for all-powerful and all-knowing, one might look to human creations. Humans have created incredibly powerful machines, that are arguably more powerful than we are, and while we are able to "pull the plug" on the machines (for now), we still don't have perfect control or knowledge of the continued internal workings of those machines (various parts may fail without our knowledge, and we have limited ability to fix, mitigate or prevent that, and we may not fully understand why a machine behaves in a certain way). Given that humans aren't all-powerful and all-knowing with respect to their creation, it stands to reason that a god needn't be all-powerful and all-knowing with respect to their creation.
It might also be that both god and the universe were products of natural processes, in which case any properties of a creator would be irrelevant.
A god that isn't all-powerful or all-knowing can still be a personal god, as they only need to be able to interact with certain parts of reality in order to be such a god.