How exactly does this example qualify as "relatively small size"?

You appear to be taking a completely arbitrary sequence and asking if it's equidistributed with respect to some measure. Clearly, unless you know more about the sequence $Frob_p$, there is no reason why it would have to have any nice distribution.

The quantity you're calling $L(\pi, s)$ looks slightly strange; why do you want to consider this particular series? It is not, as far as I can see, the $L$-function of any automorphic representation (so the notation is a bit misleading, it is not clear what $\pi$ is supposed to refer to). You might perhaps be able to write it as a ratio of automorphic L-functions.

I wouldn't call (2) a "factorisation" per se, because $L(\pi_E, 1/2)$ is outside the region of convergence of the Euler product formula. So you shouldn't think of (2) purely as a product of local terms: you need the analytic continuation of the L-function to make sense of it.

@sdr Very good point; the assumptions on $\bar{M}$ are clearly much too weak, but there is a specific good $\bar{M}$ which I want to apply this to. I've edited the question.

Take any infinite extension such that $Aut(L / K)$ is trivial (these exist). Then the condition is satisfied for all $p$ at once, but $Gal(M / K)$ is an infinite pro-finite group so it can't be $p$-adic analytic for several $p$ simultaneously.

You might also be interested in work of Ertl–Niziol which considers the relation between different versions of syntomic cohomology (with and without the special fibre included in the log-structure, so corresponding to $H^1_{\mathrm{g}}$ vs $H^1_f$ resp.).

You are correct that there is a gap in the logic of Corollary 6.8 of [Bes00], since the statement is claimed without any properness condition, but the work of Niziol cited as a reference only considers the proper case. I have been told by Niziol that the gap can be fixed by using log-structures (assuming $X$ has a compactification with boundary that is a relative SNC divisor over $\mathcal{O}_K$) but I do not know if there is a published account of this.

(typo in first comment: "did not stand still" should be "stood still".)

Your two example citations are (a) a set of introductory lecture notes which deliberately uses non-cutting-edge methods for simplicity, and (b) a paper somewhat outside the mainstream of research in this area. For instance, Kisin's work on potentially semistable deformation rings – which relies on Breuil-Kisin modules to go well beyond the Fontaine--Laffaille range – has been fundamental to virtually every major advance in modularity lifting since it was written (in 2007).