If the PGA Tour had allowed members to go play in LIV events there would be many other rival tours springing up, further diluting the PGA Tour product. The TV contracts and event sponsorship deals all have strength of field clauses the tour has to maintain to keep the money flowing, so there was never going to be a world where the tour said yes to players asking for permission to play LIV events.
If those legal challenges go through, then yes, we absolutely could end up in a world where PGA has to say yes to allowing competition in other tours.
Doesn't sound like it's their mission would be harmed by competing tournaments. After all, the majors aren't operated by the PGA and they co-exist just fine.
The PGA Tour mission statement is very different: "By showcasing golf's greatest players, we engage, inspire and positively impact our fans, partners and communities worldwide."
Hard to showcase the greatest golfers if they are off playing in different events.
The tour normally grants requests to play in other events, like world players wanting to compete in their national open or host a charity tournament. Lots of European players maintain status on both PGA Tour and DP World Tour (formally the European Tour) and a few guys have won both money lists in the a single year. There are plenty of co-sanctioned events between both tours, even some not run by the tours like the majors. The difference here is an organized league of multiple events wants to poach players without any formal agreement with the tour about strength of field and cross-competing. Suspending people who play in LIV events was really the only option the tour had.