[DevTools] browser extension: improve script injection logic (#26492)

## Summary

- Drop extension support for Chrome / Edge <v102 since they have less
than 0.1% usage ([see data](https://caniuse.com/usage-table))
- Improve script injection logic when possible so that the scripts
injected by the extension are no longer shown in Network (which caused a
lot of confusion in the past)

## How did you test this change?

Built and tested locally, works as usual on Firefox.

For Chrome/Edge

**Before:**
Scripts shown in Network tab
<img width="1279" alt="Untitled 2"
src="https://user-images.githubusercontent.com/1001890/228074363-1d00d503-d4b5-4339-8dd6-fd0467e36e3e.png">

**After:**
No scripts shown
<img width="1329" alt="image"
src="https://user-images.githubusercontent.com/1001890/228074596-2084722b-bf3c-495e-a852-15f122233155.png">

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Co-authored-by: Ruslan Lesiutin <rdlesyutin@gmail.com>