All mastodon clients I have seen involved logging in to the foreign server and/or copy/pasting handles. If this can just solve that it already removes critical friction.
Mastodon team should have made own browser extensions a long time ago, now the community is working to make this easier. I use one called StreetPass [1], but it still is a bit barebones, and doesn't yet help with following accounts accross servers. It does make discovering accounts nice.
What you bring up is a big one. Another big one for me is search. I used Twitter search all the time. For local things like "what're those sirens I hear in my neighborhood", to cultural zeitgeist like "who is this artist performing on SNL", to using it as a quasi-Google.
That's the problem here: there have been scraping efforts before, but they always receive a tidal wave of community backlash.
For a more thorough explanation
https://www.secjuice.com/mastodon-child-porn-pedophiles/
And for a github discussion where some of the Mastodon devs and hosts discussed this at length
I usually follow people based on them mentioning their handle on podcasts, or linking to their feed from their profile on another website.
The official Mastodon client accepts a web URL in the search box, to a profile on another server.
'Broken' is an understatement. It is defective 'by design' and eternally a problem that they can never solve. It is not early days either.
Hence why Twitter is still unmatched with the 220M+ daily active users.