epistemic status: I suspect these thoughts are not novel and are potentially wrong. I still think there's some value in recording my thought process
tl;dr: Selection effects from not being annihilated by aliens may explain why we don't see any very advanced aliens, but it doesn't explain why don't see any as advanced as we are. This may suggest a filter behind us.
Civilisations may sometimes become sufficiently advanced to spread across the stars. Perhaps they would even expand outwards in a near light-speed "shockwave" that converts all available resources into something that they value (like copies of their mind). Perhaps they send seed ships across intergalactic voids. Wherever they go, no new civilisations arise: all matter and energy is used by their civilisation.
Suppose we live in a universe of infinite spatial extent, and that there's some chance of intergalactic civilisation arising in any unoccupied spatiotemporal region. If these civilisations spread out at near light-speed and the chance of civilisational genesis is less than a certainty, there will always be an infinite extent of uncolonised space throughout the history of this universe. The picture below shows an example of such a region in a world where civilisations arise relatively often.
After a little thought, it might appear that we have an anthropic solution to the Fermi Paradox here: of course we don't observe any aliens, for if if we were in their future light cone, the matter we are made of would be used for something else, so we wouldn't exist.
Before we rush ahead with this line of thinking, let's look at a couple of caveats with this idea. The first is that civilisations may exist in states that are observable but not subsuming. I'll specifically consider the case where civilisations start of as observable for a period, then become subsuming. The second is the observed age of the Universe and its implications for this argument.
Visible but not subsuming
On the first, our civilisation has probably been observable for awhile, given the amount and structure of electromagnetic radiation we produce. If we presume that this "young" period is typical for civilisations, we have to explain why we don't observe any young civilisations that wouldn't have subsumed us yet.
Let's take a model where a young civilisation deterministically becomes an intergalactic civilisation over some period τ. Then we have to wonder why there are no young civilisations in the red loveheart region of the picture below.
The volume of a past light cone at time t scales like t⁴, and if p is the probability that a civilisation arises in a volume of spacetime, the probability that a light cone contains no civilisations scales like (1 - p)^(t⁴). This of course drops off very fast with increasing t.
The probability, B, that there are no civilisations arising in the red region looks like (1 - p)^(t⁴ - (t - τ)⁴).
Perhaps we think we are relatively close to becoming intergalactic (even up to a million years away would count for my purposes). That suggests that τ is very much smaller than the period of the Universe's history in which life might have arisen. Therefore, we might kill off terms in the exponent of B that are in higher powers of τ. This gives us a probability that scales like (1 - p)^(τ⋅t³). This still scales very fast with t, so p still can't be very great, but given low τ, higher values of p become plausible.
Young in an old world
Now to the age of the Universe. For a young civilisation to come into existence in some region, it has to be the case that no mature civilisations exist in that region's past light cone and that the appropriate conditions are met for civilisation to arise.
The chance that a spatiotemporal region's past light cone doesn't include mature civilisations declines with that region's position in time (very quickly, as seen above): later regions have larger past light cones that we require no civilisations to have existed within. If we assume that the chance of a civilisation arising in a region does not increase throughout time, the probability a young civilisation observes a young universe is far greater than that it observes an old universe. Our infinite universe argument says that there are young-old civilisation-universe combinations, but that young-young is more probable.
Given that it seems plausible that the Universe could have sustained life for a long time before we came around, this pushes us back towards something like a Fermi paradox. Not: "where is everyone?", but: "(why) are we late to the party?"
A filter behind
This can be resolved with a Rare Earth hypothesis, just like the normal Fermi paradox. That is, we could say it's very difficult to become an intergalactic civilisation. Even though that doesn't change the fact that early intergalactic civilisations are more probable, if it really is astoundingly difficult to become intergalactic, the probability of observing an old(ish) Universe is not appreciably less than observing a young one.
One might even be tempted (insufficiently, according to me) to say that young civilisations are very common, but that all of them die out. That way, you don't need to worry about expansion-shockwaves in your past light cone. In fact, pretty much the same proportion of space will be available for new civilisations for all time.
I don't think the last paragraph holds. Consider again the picture with the blue teardrop and red loveheart regions. Whatever the normal lifetime of a young civilisation (even if it inevitably never becomes old), there is some red region where young civilisations would be observed.
Let's model becoming an intergalactic civilisation as a two step process. Becoming a young civilisation with probability y and (after time τ), going from young to intergalactic with probability i. The probability of an intergalactic civilisation arising in an appropriate region then is c = y⋅i.
The age of the Universe suggests c is low. So at least one of y and i must be low. If i were very low and y high, then we might expect a teeming red region -- we would observe a lot of young civilisations. So that suggests that (at least) y is low and that there is a Great Filter is behind us.
The argument above is essentially using the self-sampling assumption. We consider a world with various young civilisations, and imagine we are drawn from the set of these young civilisations. What is the probability that we see empty skies? What is the probability that we see the Universe is old?
How does the above argument need to be changed under the self-indication assumption? I'm not that confident in this bit, but here's my thinking: if the possible worlds we select from are obtained by varying the parameters i and y in the above model, we expect y to be high (as Katja Grace has pointed out) because that results in more observers that we could be chosen from. The only other way to increase the probability of empty skies (in this model) is to shrink down τ, thus reducing the volume of the red region and suggesting that the time from transmitting signals to judgment day is short.