It's one option. Not the one best supported by the evidence.
And there are some problems with an eternal physical universe. It seems to require that the universe be cyclic, because if the universe is tending to some end-state (say, maximum entropy), it would already have reached that end state infinitely long ago.
I think the problem is we don't really have a unified theory yet. Loop quantum gravity, which shows some promise, posits that the universe was contracting before it expanded and that it never reached the point of a singularity. It got to a certain size and then "bounced" and at the bounce the entropy resets. From what I've read and watched, it does not say that the universe will necessarily contract again. I am not a physicist and am only an interested layman. So I could be completely off and feel free to correct me. I'm speaking from the standpoint of philosophy.
I think also a distinction needs to be made between "eternal" and "infinite". Eternal, as I inform it, would mean existing for all of time. So I think the universe could be finite and eternal, i.e. there was no time before anything existed and there will be no time when nothing exists. So even if time were finite, there would have been something at time zero. From my understanding of the BBT or now as some refer to it, the Big Bounce Theory, this is what is posited. The Big Bang Theory begins with the expansion of the universe and the singularity is sort of a place holder. Now with Loop Quantum Gravity, we have a theory of what existed before the expansion, i.e, a "planck star". But please correct me if I'm wrong.
Philosophically speaking, infinite causal chains are also a little messy.
Agreed. There had to be something to cause the first action and that something had to be uncaused, i.e. eternal.
Yes, but you can still have A being the cause of the existence of B.
Yes but if you point to something that existed to explain existence, then you haven't made any progress in explaining existence. That's why the question what caused existence* is fallaciously complex.
* I use the concept "existence" to denote the sum total of everything that exists. It is synonymous with "universe" as I use it. "nature" is existence as seen from a certain perspective: the realm of entities and their attributes acting and interacting according to natural laws and their natures.
Well that's it. Off to bed.