Nowadays are you just paying for the customer service with a premium ISP? or does the network actually perform better?
Openreach only control the access network, the tiny section between your home and exchange.
FTTC saw no contention for the vast majority of users between the FTTC DSLAM and the exchange. The rest was down to the ISP.
FTTP is the same. It's extremely unlikely you will see any contention/slow downs caused by the Openreach part of the link.
As others have pointed out the ISP controls everything beyond the exchange.
Whether you are connecting to a server/website 20 miles away, 500 miles away or 5000 miles away it will all go through your ISP. If the ISP are a bottleneck it could affect all traffic.
Your performance will also depend on what transit/peering agreements your ISP has in place for when you leave their network to the wider internet.
A lot of people assume that because it's full fibre then that must mean full speed 24/7 but that's a massive misconception.
Home broadband is a contended product. It's a shared, best efforts endeavour.
The only difference is that the connection speed doesn't degrade/decrease as distance to the exchange increases and latency and reliability should be a bit better.