I would try the FireStick in every port that you’ve had an issue with the Blu-ray player just to rule out the TV.
I would next buy a new HDMI cable just to rule that out - you don’t need to spend big bucks on that, it can be gotten for $10ish.
If you try the FireStick in every port and have no...
Chances are with the TV that one individual port would go rather than all of them at once. It’s not impossible that they’d all go at the same time but I’m not sure it’s likely.
Do you have another HDMI input device you could plug into the TV? A cable box, a streaming box, a different disc player?
It sounds to me like it’s an HDMI handshake issue - something is losing the connection and it’s then resyncing. It sounds like you’ve ruled out the cables and the TV. It could be that the HDMI board on your player just went before it’s time.
If you haven’t already, just to rule it out, I’d try...