There is nothing wrong with internet connection--that is only one of the possibilities why the repository isn't available. The real problem is found lower down. It is nothing to do with browser, but with settings for which repositories should be used. They're stored in text files at /etc/apt/sources.list, and under /etc/apt/sources.list.d/ BUT you can fully access them from Synaptic.
Synaptic, Apt, and apt-get are all the same software, loading .deb repositories, just with different interfaces. If one has a problem, all will have the same problem.
The Brave issue is that Brave no longer issues an i386 version. I can't say why your sources.list specifies one, but it doesn't matter. You don't need to reinstall all the software, you need corrections to your file that tells synaptic/apt where to find the software packages.
You can do everything about the Brave you need in a few seconds, within Synaptic, on the repositories lines. Just change to match what is on the link reference on the brave forum. Then reload repositories, then update. In Synaptic, choose Settings > Repositories. Then Other Software tab, and highlight the repository affecting Brave. Make the line look like the sample linked after the deb part -- meaning add the [arch=amd64] and that should be it. Or, you may have some reference locking into i386 in the same line that needs deleting. Close, reload repositories, update and you should be good.
The secure error is simply adding [Trusted=yes] between deb and http on the appropriate repository source line--also can be done right in Synaptic.
Or better, and not bypassing security, get the proper keys to the repository as a key file, and on the same page in Synaptic, two tabs to the right is the Authentication. Import the downloaded key file, and that's it. This issue was likely caused by manually entering a repository without importing a key file at the same time.
If the repository can't be troubled to provide a key file, I would be seriously questioning whether I need that repository, and whether it can be trusted, because it is trivial for them to do so.
Lastly, apt-secure is a basic component of Ubuntu. It is in the Apt package, which is required to have Synaptic running. I see the post was edited, so I suspect you figured out not to type the $ (or #) when someone offers a command to type? The $ or # is merely indicating whether you need to run the command as regular user or as superuser.
So there's no reason to go back to command line, nor expect to spend hours. All of us are busy, including people who take time to write answers on forums 