trying to restore 3rd party app from time machine

Not all third-party applications simply install into your Applications folder. Some spew other dependencies elsewhere in the filesystem (e.g. /Library, ~/Library) that are not restored by Time Machine when retrieving a specific application. That may lead to partial or full application failure and requires reinstallation of that specific application, where version dependencies are critical.

This article from Apple for Restiring Items from a Time Machine Backup Drive

Restore items backed up with Time Machine on Mac

Though, this does get the question

" However, my Time Machine only lets me go back one day (not sure why- I've never had to use it before) and so it "restored" the current version."

Normally, when Time Machine Backup has been setup, it works on a Schedule

Learn about backup frequency and duration

Now, predicated on you TM Backup was setup like this AND the Destination Drive is connected to the computer

TM Backup will backup on schedule without your Interventions

Now, is as you have written " I originally got it in 2022 " - 4 years ago

It is possible your Time Machine Backup does not go back 4 Years

Sometimes you have to wait a long time (minutes) for Time Machine to reveal more than a couple of recent days. If you know you have more backed up dates than TM shows you then go into the TM and give it time (make a cup of tea, do the washing up). If that doesn't work then reboot the Mac and try again.

The old version of the App should show up in the App Store > Account > Purchased on you Mac.

What third-party App...?

When and were did you originally acquire that App...?