Paul Romer's mathiness paper: a year later (1)

For this first post of the (recalling) Paul Romer mathiness series, Appendix B of https://paulromer.net/wp-content/uploads/2015/05/Mathiness-Appendix-Expanded.pdf is the main topic.
(Paul Romer blog post: https://paulromer.net/mathiness/)

tl;dr version: Paul Romer is right, but Lucas and Moll aren't really wrong also. But things to think about are there.

The paper came to my mind, as I was playing with my brain spare time capacity, while I was (and still partly am) suffering from feeling sick. (not a good way to live a life - it's not that I am like this for other times, but sometimes, you are struck with some trains of thoughts for no reason) More specifically, I was thinking of use of double limits, then I was reminded of the paper.

Since there'll be people not wanting to go alternating between the paper and this post, I will write down some of Romer's core arguments using Romer's notation. \(\beta\) is the innovation arrival frequency. Now consider the following results:
$$\lim_{t \rightarrow \infty} g_{\beta}(t) = 2\%$$
$$\lim_{\beta \rightarrow 0} g_{\beta}(T) = 0$$
\(g_{\beta}(t)\) is the economy growth rate given \(\beta\) at time \(t\). The second result holds for any \(t=T\). In this circumstance, which one of the above two equations is the right way to think of long-run growth rate? For more clarity, change the above two equations to:
$$\lim_{\beta \to 0}\lim_{t \rightarrow \infty} g_{\beta}(t) = 2\%$$
$$\lim_{t \to \infty}\lim_{\beta \rightarrow 0} g_{\beta}(t) = 0$$
The first equation choice is what Bob Lucas and Benjamin Moll used in their paper. And without (and possibly even after) reading Paul Romer's paper and appendix, many would feel that this is a justifiable choice. The second limit says: suppose that innovation arrival rate is close to zero. Then let's see what happens as one goes long-run. The first limit says: let's first move forward to long-run and see what happens.
As long as \(\beta\) isn't close to zero, the first limit is the sensible one, right?
But one can think differently. After all, it is generally assumed that we cannot really "see" infinite time, and all times are finite. If the so-called long-run growth rate is not achieved - and actual growth rate remains below the so-called long-run growth rate (the first limit one) - after a short period of time, but rather requires so much time, possibly "almost never" (because \(\beta\) is so close to zero), what is the point of talking about growth rate? Growth rate is about growth happening over time, and if long-run growth rate is not what prevails for so much of time, any conclusion following from the first limit long-run growth wiould be mathematically correct, but economics-wise meaningless.
Thus, one is forced to discuss actual economic growth paths reaching up to the first limit long-run growth rate rather than just abstracting away these processes and buidling up conclusions from the first limit long-run growth rate, says Romer or Romer per my understanding.

(side question: while the first limit long-run growth rate is "unobservable", would Romer's statements about this unobservability defeating the theory really be justified? [talking about the appendix here too.])

Now reminded of this paper, I actually was reminded of my own thought a year (I believe?) ago regarding limit. It is not exactly about the double limit issue above, but quite a similar one. And I plan to write about this after this series.