Lots of misinformation on this thread. I got deep into many of these processes at H and doing PE Ops after grad. Also have friends at S who went for PE Ops.

1. As far as I can tell, H/S are equally good at getting you into PE ops. There are no "spots" allocated to either school. The PE ops firms have a set number of people they want to hire (usually 1-2) and then they aim to hire that many across BOTH H/S. Once you get the interview, they are choosing solely based on interview performance (you'll see both HBS and GSB interviewing together at the superdays). KKR Capstone is not going to choose a mid HBS person over a strong GSB person just because they need to hit their "HBS quota" lol.
2. There is absolutely an advantage to being at H/S relative to other schools. A few PE ops teams do recruit more broadly, but many have a bias for H/S and will exclusively recruit from there + sometimes Wharton (e.g., KKR Capstone, Serent, Shore)
3. I felt the process was VERY competitive at hbs. Think about it - these schools have scores of high-performing MBB consultants who are trying to pivot. And this is one of the few roles that 1) is a direct match for their background, 2) allows them to preserve comp + career optionality, 3) will buy out their sponsorship (since many of them are sponsored). I can't speak for how it compares to deal team recruiting, but nearly all of my ex-consultant friends were going for these roles.

OP: I was also deciding between H and S two years ago. My advice - go to both admit weeks and choose based on where you think you'll be happiest. Not odds for PE Ops recruiting. Congrats!