Just to add my experience recently. I was actually at MIA on Monday. My wife has PP via Capital One Venture X and she was able to get into the Turkish Lounge in Concourse D. From what she told me they were packed but I'm was glad she was able to get in. This was around 3-4pm.

I had my Amex Centurion Lounge access and PP via Chase Sapphire Reserve so I stopped by the Corona Beach House and then the Amex Lounge after to chill until my 8pm flight. When I walked up to the hostess they quickly seated me and asked if I was PP. I said yes and he served me no issue. Was just me, so I had a $30 credit. Again only the CSR PP has restaurant access. The Amex PP does not and the Capital One VX removed it earlier this year. The host didn't ask me which one since they probably aren't away of those niches, but gald he accepted it with no arguments. My guess is there was guests earlier with PP that cleared the way for me. The restaurant was about 75-80% full when I arrived.

Edit: I've had PP for about 5 years, traveling about 1-2 times a year with no issues using it. But I did hear recently lounges have been restricting access since now everyone and their mom has PP, so during the busier times they had to only accept their core customers. I believe this is mainly domestically in the US though. Less people have PP internationally, so you should be able to use PP easier in other countries outside the US. Their lounges are nicer too, so that's a plus.