But at 5 years old you are reaching the design life of these things. Did you try setting the BD players to say 720p output?
That 1080i could be an issue on some players. And you may have proved it since your older un-named player did work.
I have an older 1080i plasma tv (2007) and a new LCD
I just bought a samsung and a sony blu ray.
both blurays have picture noise on the plasma but not on the LCD the resolution is right and both have been updated.
My old blu rays a panisonic and a visio did not have this problem so long as the resoution was right on the bluray.
What can be done the tv works great but dvds work better than blurays. with the new players.