I can't imagine hiring a freelancer that I felt I needed to micromanage to that extent. What do you do when you need to spend some time working on a white board? I probably spend more time on a whiteboard than I do coding (I'm more of an architect these days, but that was still true when I was only an individual contributor.)

I worked as a freelancer for years, and I would have refused to work for someone that required me to install tracking software.

If it's possible at all, I would work on trying to bill daily or weekly instead of hourly if I were you.

There was an option to post "manual" time without screenshots; I have used it a few times, but mostly I was sitting before the computer anyway. I doubt most of my customers actually looked at the screenshots; I remember one distinct case when a customer actually did, read my todo list for the project and provided feedback and answered questions without me even asking :) It was when we've just started working though, so naturally he was cautious.

Also, I my case it was a requirement from the site (Upwork); in return they guaranteed the payment, that is, if I provide screenshots, I get paid by Upwork even if the customer doesn't pay. When I started a few years ago it was very reassuring, although I don't think this actually happened with my customers.

One only needs to google the name "Upwork" to find out whether they indeed "guarantee payment".

They don't. Not by a long shot.