Installed and tested it with Geary
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A lot of work to do here. So far this looks like a copy-pasta from Arch and it shows in the final result. Even though it's a derivative work, we treat this as plagiarism and ask that you not do this, especially without any sort of attribution. Please take the time to address all of the inline comments, as well as ensuring that you understand everything that is being done.
So, I don't really see a reason to use the git support when they aren't actually using Git submodules. You should be using versioned tarballs instead. And given that they seem to be tagging the bridge and the GUI separately after the 1.3 series, they'll need to be two separate packages.
Should be using pkgconfigs where possible: pkgconfig(libsecret-1)
Same goes for the Qt deps. You can see these in the Provides section of eopkg info when looking at the -devel package.
You can use install -d without specifying access permissions because all directories are 00755 by default on Linux.
This isn't needed since -D takes care of it. You only need to create directories that you are copying or symlinking into.
If you are copying, you need to go back and make sure the permissions are set to read-only, with execute where needed.
Seems like this should just be proton-bridge without the .git. Though I'm guessing if you were building against the system Qt5, you could just install the binary to /usr/bin and forego /usr/lib64/protonmail-bridge altogether.
SVG icon should be named protonmail-bridge.svg and go in: /usr/share/icons/hicolor/scalable/apps/
We don't copy licenses unless the build system does it automatically. It's already documented in the metadata.
Guessing you didn't rebuild after fixing these.
This needs to be using the system Qt5, vendoring is not acceptable.