Let's go over the circularity issue in more detail. It starts with Swebs' statement that "[The big bang as a uniform expansion in spacetime] wouldn't work out because it wouldn't match what our telescopes are observing." Here we have a claim that the 'uniform expansion in spacetime' hypothesis (UEIS for short), which we all agree is wrong, has already been ruled out by astronomical observation.
Next, Swebs offers a specific argument for that claim, in the form of a diagram which shows that, in a uniformly expanding universe, each point appears to be that from which the expansion is ocurrring. One problem with that argument, which we both apparently agree on, is that this would be true for UEIS (for one thing, the image can be taken as a diagram of a demonstration using marbles on a table, which would literally be a UEIS.)
There is a second problem with it, however: none of our telescopes have observed the universe from the perspective of a distant galaxy, so we cannot say that our telescopes have shown that everywhere seems to be at the center of the expansion. We can deduce what it looks like from distant galaxies, if and only if we make some assumptions about the dynamics of the universe (there may be other observations from Earth that have already ruled out UEIS empirically, in which case the rest of this comment is moot. You have claimed that there are, but so far, you have have not presented any details.)
There are two possibilities: either it looks the same as from Earth (putting aside details local to the specific point of observation) or there is some difference. In the former case, which would hold if the expansion we observe is a uniform expansion of spacetime (UEOS), and which therefore is what we all, and cosmology in general, assumes to be so, I do not think these observations would rule out the possibility of UEIS. Therefore, for observations made in a distant galaxy to empirically settle the UEIS/UEOS issue, they must differ from those from Earth, in a way that is diagnostic of, and therefore causally dependent on, what form the expansion takes. Furthermore, because the local and remote results would differ in this case, UEOS would be ruled out.
So, when we try to deduce what can be observed from a distant galaxy, we must either assume that it is the same as from Earth (subject to local corrections), which will fail to resolve the issue, or we must deduce that there is some difference - but what difference? As shown in the previous paragraph, the relevant differences must be causally dependent on what form the expansion takes, so you have to make assumptions about the latter in order to deduce the former. There's the circularity, and it is general - i.e. not dependent on what specific measurements are being considered.