I did that once. In very polite terms I told them that I like to keep personal and work activity separate as much as possible for personal and work security reasons and that if they issued me a phone with an app I'd be happy to carry it.
I got a very positive response. Ultimately they didn't think it was worth issuing me a phone and everyone went on happily.
It’s increasingly common for employers to give payslips via a smartphone app; which needs to be installed on a personal phone of course.
I’m one of the very few people to make a stink about it. But they’ve accommodated me after a lot of back and forth.
Playing the devils advocate though: installing an app is not necessarily super invasive, especially if the app don't get any permissions (and / or is one of the apps that are ok with getting installed in for instance an android studio-style android emulator).
That said, for sure it would be super nice to be able to get a "real" information on what tracking things happen (unfortunately that's not realistic) and if there are requirements from the company for me to use the system (that would theoretically rely on the install of an android app for instance) I would personally require that I either get a company cell phone to install them on, or some sufficient guarantees (on the level of physical paper contracts signed signed by the head of HR) of what they will do.
Or if they would accept me installing the app (and their app manifests etc) on an android vm on my laptop (like you can get in android studio), in that case I guess it would maybe be kind of acceptable to do it (at least on the work laptop).
The limit I will personally go to on my personal devices is installing a (generic, unrelated to what the company asks me to do) TOTP app and registering 2fa auth in that app.
I personally have never had a request for a company-provided work phone/tablet denied, if they required me to do anything on my personal phone that was critical for work.
If they ask why, I tell them I have a BlackBerry :) never been a problem
What was an interesting discovery was that PingID doesn't require a cellular data connection to work, so I was able to use the phone unregistered (so no network charge) and permanently in flight mode (to preserve the battery).
Even now after leaving that company I still haven't registered the phone or taken it off flight - I just connect to a personal wi-fi as necessary.
I don't blame BigCorp for their policies; their equipment, their rules. But I strongly recommend separating the use of business and personal devices.
And, no, if an employer demanded I install an app on my personal phone, I'd refuse.