Interesting question.
As a classical theist, I never took polytheism (at least not of the ancient Greek pantheon sort) seriously. For one, it is such a metaphysically extravagant view that it just cannot be affirmed on the evidence we have without violating Ockham's razor. Other than that, the problem of utter absence of uniformity or moral excellence of this pantheon is basically a Pandora's box of problems with regards to divine-human relationships, obvious antropomorphism and even global skepticism.
Your question illustrates another problem with the view and that is the question of the pantheon's competence and/or desires. If we are to grant that these gods actually exist (as I take your thread to be a question of what kind of picture the decline of polytheistic religions paints of said gods rather than the question of those gods actually existing) then we are faced with two, equally disturbing alternatives:
1) the gods were unable to prevent the doom of their religions
2) the gods were uninterested in the doom of their religions
On (1), the gods wanted to maintain their religions, did what they could and (for whatever reason) they failed. As such, they end up being rather incompetent and fall short of the superiority over mere mortals that is so often attributed to them. It makes little sense to seek connection with or worship these beings as they are apparently unable to sustain such a relationship (if they can even muster enough power to start it again).
On (2), the gods had it in their power to maintain their religions and could have easily (for all we know) shattered the plans of mere mortals, but it happens to be the case that they are simply not interested in being worshiped and judged it as fitting that no mortal be bothering them with their cries. While this does nothing to damage the cosmic standing of these gods, it does conclusively abolish all need for polytheistic religions in question. If the gods do not want to be worshiped or are (at best) apathetic to it, what reason is there to worship them?
Both options, which I (funny enough) saw a couple of polytheists here actually endorse, take away from the rationality of polytheistic worship. So, does the doom of those polytheistic religions necessitate a rational person to stay away from them? Despite my strong anti-polytheistic sentiment, I do not think so. Mere demise of these religions at one time need not mean said religions cannot be revisited. I think a third, more satisfactory (at least as satisfactory as polytheism can get) alternative can be offered and it goes as follows:
3) the gods were able to prevent the doom of their religions and found it appealing, but a greater cause compelled them not to intervene
To elaborate, it could be the case that Odin, Zeus, Ra etc. wished to keep their religions alive, but realized that it will be for the greater good to have those religions die. One good possible reason could be that the gods saw (through whatever magical means) that their religions will be revived later in the distant future and be so strong that they will be able to definitively destroy all their oppressors, thereby making polytheism ultimately victorious.
On this view, polytheists seem to, at the very least, not be completely irrational in seeking connections with their gods for them doing so may be the start of the great revival of their religions. As such, it doesn't strictly follow from the death of those religions that attempting to engage in them again is irrational. This is, however, mere speculation, one that may be enough to constitute a broadly logically possible solution, but not one that is anywhere near obvious or supported by evidence. It also leaves open the question of how to fit this view with traditional beliefs regarding those gods, so the polytheist has a lot of work to do if they wish to rationally hold their polytheism.
The objection, while definitively an interesting one which tells us a lot about the poor state these gods are in, is not one that can establish the irrationality of polytheism. It can, however, be a part of a cumulative case against polytheism which is, I think, more than capable of accounting for why polytheism is a far inferior view than monotheism.