And one more update, hopefully the final one.
Even the wifi extender I mentioned in my last post did not solve the problem. While the signal was very strong (30-35 dBm), the Firestick was still losing connection very frequently. So, I did some more research and found some useful info on the web. Apparently, Firestick only supports channels 36-48 on 5GHz, and if the router is set to auto, it seems that anytime the channel is not within this range, Firestick will automatically try to switch to 2.4 GHz, and go back to 5GHz when the channel is back to 36-48 range.
So, the recommendation was to change the 5GHz channel, from auto to anything between 36-48, and also rename the SSID of the 2.4GHz network to something different.
This whole thing did not make much sense to me (especially renaming the 2.4GHz SSID), but I decided to give it a try. It was working fine for a few days, so I returned the wifi extender, previously purchased. This was all about a month ago, or so. My Firestick still works fine on 5GHz network. Hopefully, that's it.
I still can't figure out why the Firestick worked without a flaw, a few months ago, when it was much closer to the router, without me having to change the router settings. The problems only started when I had to move it to a room with somewhat weaker wifi coverage.
I'm posting this here in hopes that this info will be useful to anybody having the same connectivity issue. Changing the 5GHz channel to 36-44, instead of auto may be the solution.