I have exactly the same issue, stuttering picture, sound is fine. All used to be fine, only stated few months ago. BT Sport is fine on iPad, stutters on Imac. Download speed is fine, got the latest router, BT must have changed something. BT Sport is unwatchable on Imac, very frustrating
A further update......I am also still having this issue. I do believe the player is part of the problem. It seems to be very poor at coping with anything less than a perfect connection. After much investigation they believe there is an issue with the copper connection between the cabinet and my house. Even though i am mostly getting good speed I am having some packet loss and this it would seem that this is being reflected in the sport app on the computer. I am actually getting great support from BT's tech team but as yet the problem has not been resolved, there is talk of laying a new copper cable but I am suggesting that if they are going to dig up the road they should connect the fibre into the properties. I probably wont get that but there is no harm in asking. I would suggest taking a look at the logs generated by your router to see if that gives any more info and also being very pushy with BT in terms of getting support. I will keep you posted.
I have a mid 2010 iMac running High Sierra 10.13.6 and have been having similar problems with the picture freezing etc. This was happening on both Firefox and Safari. After trying various things without success the BT technician suggested trying the google chrome browser, and happy to say it works perfectly now.
I know your post was some time ago now , but hope this helps.
It's been a while but I thought I should give an update because things have improved. I doubt this is an improvement introduced by BT per se.
To recap, BT sport on my 2008 MacBook Pro has been really poor. Unwatchable, in fact. Out of curiosity, I tried it again last week (an FA Cup game if I remember, but don't remember which). I started watching the Enhanced Player. Sure enough, just as before, it started at low resolution (really blurry) but no stuttering. After a few seconds it switched to a higher res, and then started stuttering. This was exactly as before.
I then clicked the teal menu in the top right of the player and chose Line ups or whatever it is. The game switches to a mini window within the bigger window. I left it there for a while, and it continued to stutter. Then it stopped stuttering. I closed the team line up thing and the game window went back to normal size. I had no stuttering from then on. I'd have to say that the resolution was a bit lower than the high res at the start but it was perfectly watchable. I didn't use any of the "enhanced" features of the player.
I can only conclude from this that when I first start the player it has a problem working out what rate/resolution it can support (if it is indeed rate-adaptive). And it continues to fail to work it out. Something about switching to the mini window seems to allow it to work it out. Once it's worked it out, it's fine.
FYI, this is in Chrome on High Sierra 10.13.6 (which is offically unsupported on my MacBook Pro - but installable using the method described in dosdude1.com/highsierra). I haven't tried Safari.