Any modern IDE has those functionalities, and most of them run on Linux too. I don't think there is anything you have shown that warrants sticking to Tortoise Git instead of any other IDE.
Also if you don't want to use an IDE Gnome's file manager nautilus supports scripts available on right click. This would allows you to run git commands from the file manager , show the information you need using [YAD](https://yad-guide.ingk.se/#_introduction)
Tortoisegit isn't an ide though, and the quality of git integration varies wildly between different ides - many of them have pretty toy integration. Its much nicer to have proper quality support of uniform quality, rather than having to rely on whatever you happen to be using at the time
Ideally I'd like to avoid having to reimplement the whole thing from scratch using ad-hoc scripts
Also if you don't want to use an IDE Gnome's file manager nautilus supports scripts available on right click. This would allows you to run git commands from the file manager , show the information you need using [YAD](https://yad-guide.ingk.se/#_introduction)