Something's definitely going on with picture quality on Sky Sports with Now tv. About five years ago when I first took out a BT subscription I had the option of the Sky Sports 1 and 2 add-on. I took these and there were no problems with the pucture (same tv). But they were 'proper' channels in the sense that you could record them if you wanted. Not possible to do that now as Sky Sports is only avaible via the Now tv app. I don't know the thinking around why BT agreed to this or maybe they assumed the picture quality wouldn't suffer.
As you say, football on Amazon doesn't have this problem so why does Now tv? I suspect you are right and something is being done to the picture at the processing stage. But why this needs to be done, and who exactly is doing it, will probably remain a mystery. Some of the Amazon games were also shown in HDR. Absolutely stunning picture. The BBC showed some games in the last few football tournament's in HLG which also looked fantastic.
Had a quick look at football in a friends house, who is a Sky subscriber, on his tv. Much better than Now.
In order of best picture quality I have seen at home;
1) BBC HLG and Amazon HDR
2) BT 4K
3) BT HD
4) ITV HD
5) BBC1 HD
Obviously, I don't know what Sky Ultimate looks like or SKY HD (apart from the quick look noted above)
