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why some softwares are encouraging people to verify, or at least point it out how, but some are not even bother mentioning anything?
Different priorities, I guess
the GPG suit is quite interesting, once you export the private key you still need the PW that you set before to import it, what would happen if you lost that? but isn't the private key like the seeds which could be used independently , or is it because I'm using a third party tool so I need the PW to unlock the key?
It's because your key was encrypted and the password is required for decryption. It's also recommended to set a password. So yes, it's like seeds that can be used independently—at least I would think that's how GPG suite implemented it but can't verify, since I don't have macOS. Therefore, if you lose the password, you lose the key. So use password manager to use strong passwords without forgetting them or this method:
I would still recommend using a password manager but seems like you already do
going down this verifying rabbit hole, now I wonder what's the safer way to download addons, e.g. Alby extension or Bitwarden.
Mhh, good question. I think you only have two options: use the official store to which Bitwarden themselves link to here or build the extension yourself but I imagine that's going to be pretty difficult since it's only intended for development.