What does it mean though to say time ran slower in the past than today? At its most basic time dilation is a scheme for comparing different clocks, but if we say time ran slower in the past that suggests we are comparing (or at least can compare) the rate of the same clock at different times. That’s why I say it is nonsensical.
[quick note: “co-moving” as used below means that an object/observer is in the frame where the CMBR appears isotropic, we will assume (slightly incorrectly, but not massively so) that we are co-moving.]
Red shift/blue shift (I hesitate to use the phrase “Doppler shift” as that usually has more specific connotations) is how we compare clocks by and as I said in fact in the big bang model, in an expanding Universe, faraway “co-moving” objects (and due to the finite speed of light it means we are also observing them in the past) appear to be red-shifted. Hence from a purely visual point of view if we were to observe a clock on such an object it would indeed appear to run at a slower rate than our own. It may seem then it is perfectly sensible to talk about time dilation in the past vs the present by looking at faraway clocks, but unfortunately it is not that easy.
In relativity we don’t tend to take the total red shift/blue shift factor as the time dilation factor. For example in special relativity the gamma factor which describes time dilation is taken from the relativistic correction to the Doppler factor rather than the Doppler factor itself. Hence in special relativity when a clock is travelling towards an observer it will actually be blue-shifted (i.e. visually it will appear to run faster than their own clock), however we still say that it is slowed down from the point of view of the observer compared to their own clock as it doesn’t (visually) run quite as fast as it would without the relativistic correction.
How then in our cosmological scenario do we decide which portion of the red shift is due to time dilation and which portion is due to “other factors”? Well the answer is that we apportion all of the red-shift of co-moving objects to “other factors”, the “other factors” specifically being the expansion of the Universe. The reason for this choice is lies in the coordinates we choose and that in these coordinates the cosmological time corresponds to the proper time of the class of co-moving observers. It could be rightly pointed out that spacetime coordinates are arbitrary, however some coordinates are less arbitrary than others and we have actually specifically chosen the cosmological spacetime such that it possess these spatially homogeneous and isotropic coordinates.
Now as you’ve pointed out, gravitational time dilation is an “thing” in relativity, so why doesn’t it apply in this situation? The answer is that its zone of applicability is actually quite narrow, applying to certain situations which are neat enough (or more properly, possess the right symmetries) for it to make sense as a coherent concept. It’s not that in other situations clocks all run at the same rate regardless of what gravity/spacetime is doing, it’s just they are too messy for something as nice and as neat as gravitational time dilation to emerge.
The two ingredients required for gravitational time dilation to be defined are asymptotic flatness and staticity (i.e. the property of being static).
An asymptotically flat spacetime is one in which is typically characterized by an isolated collection of mass or masses, e.g. a star in otherwise empty space. This first ingredient arguably is not absolutely essential, but what it provides is a a notion at infinity of there being no gravity so that we can compare and say “a clock runs slower here than it would at infinity where there is no gravity by this amount”.
A static spacetime is one that, if you like is strongly-independent of time or more specifically there exists coordinates where the space coordinates are independent of the time coordinate and vice versa. The reason this is needed is that the independent time coordinate can be compared to the proper time of a clock at the independent space coordinates in a way that is consistent for all clocks in the spacetime. Without this ingredient the comparison becomes arbitrary and inconsistent.
In standard cosmological models, spacetime is neither asymptotically flat or static, so gravitational time dilation is not a concept encountered.