Congrats on getting it fixed!
I work in the plumbing trade and would offer the following:
Polybutylene got its bad rap from the fittings failing, not the pipe itself. I have worked on lots of jobs with PB and as long as it's still flexible, I don't try to convince homeowners to replace it, especially if it's obvious that they can't really afford to. The class action suits happened due to flooding catastrophies caused by fittings leaking, most likely poor installation jobs. PB was not designed for interior plumbing systems. The systems used mostly home runs with minimal fittings. The story I heard was that ambitious salesmen got together and convinced plumbers to use PB for applications it wasn't designed for. The plumbers used tons of fittings, and buried them in walls, a good number did it really poorly! Back in the day, I walked up to PB failed jobs and saw water running out the front doors of apartment complexes.
I have jobs with PB that were build in the 80's. So they are going on 40yrs old! I have replaced lots of fittings and old distribution manifolds, all connected to the original PB. You will hear lots of people tell you PB is horrible, and to never buy a house with PB. I have lots of experience that proves otherwise. So I would recommend checking for structural integrity, and flexibility, then use the proper transition fittings and continue on with pex.
The transition couplings are less than a dollar to just a couple of dollars, much less than the sharkbite fittings, but yes, you do need a crimp tool. You will save money in the long run buying a crimp tool and fittings rather than sharkbite fittings. I have had sharkbite fittings leak a disturbing number of times, and never use them anymore! I know lots of plumbers who use them, they are quick and easy, but, In my experience, as well as the experience of friends in the business, they are not a safe long term fix, and should never be buried in walls or inaccessible areas.
The fitting you want is this
If you get it at Home Depot or lowes, it comes with crimp rings, the bright one goes on the PB,
And the end with closer ridges goes into the PB. They are cheaper at the plumbing supply house, you just need to buy rings separately.
The cutter you ended up will work fine, the trick not to crush the PB, even with that cutter is to put a little pressure on the pipe with the cutter, then rotate the cutter back and forth a little. This will start the cut and make it easy to finish.
This cutter from Home Depot is fine also, only $7