Limited Network Connectivity after reset and reinstall of original macOS

Very little on the Internet will work with Mountain Lion, because of expired root certificates, lack of support for more modern security protocols and reliance on security protocols that are now completely deprecated. You need to do a few things:

1. update your web browser to something more useful than an obsolete Safari. The last version of Firefox to support Mountain Lion is Firefox 45.9 ESR, which is still barely usable but should at least be enough to get you going while you upgrade macOS: https://ftp.mozilla.org/pub/firefox/releases/45.9.0esr/mac/en-US/Firefox%2045.9.0esr.dmg
2. You need to update your ISRG X1 root certificate. You can do that via Let's Encrypt: https://letsencrypt.org/certificates/. Download the self-signed ISRG Root X1 in .pem format (save it to your Downloads). Open Keychain Access, select the System keychain and then drag the .pem file to it. It will ask for your password. Then open the certificate and change the trust settings for SSL to always trust. Again it will ask for your password.
3. At this point you should be able to browse the web - and more importantly, use the App Store or the command line to get the Mojave installer so you can run the upgrade.

Your macbook air is now 8 years old.

It may be worth upgrading to a used Macbook air M1 or M2 instead of spending your time on this.

I gave up on my Macbook Air 2013 a year ago and the Macbook Air M2 I have is so much more reliable.

I was spending too much time trying to keep it working and updating the software.