Hmmm... text not deleting is probably some compatibility stuff.
I actually had several friends play it to be sure it performs ok if not on mobile and there was no such issue.
It looks like variations in browser setups/OS-s do affect Godot in nontrivial ways which are hard to detect and debug because of their setup-specific nature.
One would think that converting to webassembly/asm.js should eliminate most platform differences however it seems not to be the case and probably it even adds extra factors because of differences between webassembly implementations.
It might actually be that plain Java games are a bit more reliable, so i need to look into more minimalistic options next time