Are you aware of Redlib? Self-hostable frontend for Reddit aimed at privacy. I’ve never had a problem with old.reddit but Redlib has a bit of a more modern UI if that’s what you’re after. There are a bunch of public instances if you don’t want to host it.
Otherwise I’m sure you could use uMatrix to disable the tracking (can’t give detailed instructions sorry), but I’d argue hitting Reddit’s domain at all is already less than ideal if you’re trying not to be tracked.
Libnotify backends are D-Bus services, which isn’t really something you’d want to implement in a shell script. Going by some source code I just found, it looks pretty straightforward to do in Python, so that’s one option.
The easier option would be to use an existing notification daemon that lets you disable the default GUI and specify a script to run as a hook, but I don’t actually know of any like that.