My honest recommendation for Twitter: Don't use the built-in systems to follow anyone. Keep bookmarks on people's profiles and visit manually if you want to know what they're up to. The timeline is a terrible idea. Twitter works really well for me because of this.
tweetdeck.twitter.com[0] works really well for this. It does not "filter" your feed in any way (as far as I know). You follow a hashtag you see every tweet for that hashtag. You follow a user and you can filter out "retweets" and "likes" to only see what the user posts (I wanted to see what @POTUS tweets and it's nothing only retweets).
A column in tweetdeck can be a user, a search, a list, a hashtag, and more. Every column shows ever tweet and they mostly come in realtime.
I'm not sure I understand, what's the problem with following people and using the timeline (maybe even in a third party client that sorts it chronologically)? What do you gain by only navigating to profiles manually?
The timeline is controlled via algorithm, and either filters or selects things to show you based on an uncontrollable process which is likely just meant to make you more outraged, annoyed, etc. to keep you engaged with the platform. Not only is that unhealthy, but it doesn’t make for an enjoyable experience almost by definition.
Can't you just set the timeline to show latest tweets first to avoid this? The alternative is certainly terrible (though mostly because it hides a bunch of tweets I actually wanted to view rather than because of any outrage), but the option to disable it is there.