OK. For those who will be visiting this topic in the future:
The problematic part that made the whole Volumio unresponsive was an external plugin called miniDLNA. No need to investigate the reported issues further - I’m surre it’s the pluggin which is guilty of messing with the system; I’ll be trying to solve the issues with the plugin’s author.
After disabling the plugin for good and making some additional tests, I can happlily let you know that Volumio is snappy like it should be. Therefore my problem is solved.
The tricky part was to spot that it’s the plugin’s falut (a funny thing is that it’s the only external thing I’ve added to Volumio ever), because it doesn’t squeeze any measurable resources out of my Raspberry Pi. Hence, it must be messing with the system on some level that results in introducing a lot of delays in network connectivity / rendering the interface in the browser. Also, there’s still nothing in the logs, but I’ll try to convince the author to inspect my case.
On the other hand, if any of the Volumio developers are going to stumble upon this post I think it’s worthwhile to figure out why this popular DLNA extension makes Volumio unresponsive.
PS: My miniDLNA version is 1.1.7, but I was experiencing it impacting Volumio negatively for at least a couple of months at this point.
I also recommend all Raspberry Pi users reporting networking and performance problems of their Volumios to check if they have miniDLNA installed and enabled. There’s no way that Volumio is going to be glitching due to performance reasons on Raspi 4 - it has like 10 times more powet than Volumio’s needs.