I don't know how well this fits your usecase, but Proxigram is a selfhostable website with which you can access Instagram anonymously. I don't think it supports following accounts in any way though.

You can try it out on one of the public instances.

I did also try making an android app for it, but that app is barely even able to display posts, and as I realized that I just don't use Instagram, I didn't really bother continuing it.

If anyone here is in the mood to make that app happen, feel free to use my humble beginnings as a starting point.

instaloader can scrape instagram accounts. do note that they'll ban your account if you use it in a 'suspicious' way, i.e. downloading multiple times or simultaneously.

you can setup a cron job on your *nix machine and setup a script to notify for new files (posts).

Instagram Lite is an okay alternative to the bigger, main client and since it's official, it won't get you banned. It's not available in most markets, but sideloading from APKMirror works very well.