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.

[0] https://tweetdeck.twitter.com

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.

AFAIK no. But there exist third party websites and apps that do. Or at least used to; Twitter may have changed it since I last used TweetBot.

But if you only follow people which profiles you would want to visit manually you wouldn't see all this outrage?

You might miss other more positive or neutral messages from those people.

I don't think messages are filtered out, they are just shuffled to show you the ones first that the algorithm thinks you like best.

But if you don't check frequently enough or read long enough then it is effectively filtering out tweets that aren't near the top.