As far as I know, having a newer QT won't help with porting Chromium (the open source part of Chrome) to OS/2 porting Chromium would be a huge job.
Palemoon is a possibility as it is a fork of Mozilla. I have a very broken port here that does display web pages.
Rust is basically alpha software, or was until recently. It is built on LLVM
https://en.wikipedia.org/wiki/LLVM, so first LLVM would have to be ported. LLVM is a project of similar size as GCC and to get it to the point where it produces code, including DLLs on OS/2 by itself would be an undertaking.
Then Rust 0.01 needs to be ported, which is used to port Rust 0.02, which is used to port Rust 0.03, etc, repeat about 20 times to get to Rust 1.0, which might be able to build current Rust. Every 6 weeks there's a new release and Mozilla uses the latest.
It's a lot of work, which would involve at least one compiler expert, which we don't have. Even some of the BSDs have only lately got current Rust running, and they're starting with LLVM already ported and using ELF objects, shared libraries and executables, unlike us. We don't have a compiler expert and the funding for one would possibly be a couple of hundred thousand dollars or more.