I love the 70-200 f4L !!!
Use it whenever I can! It's lightweight! Fast focus!! Super sharp! And a bargain at the used prices of £350!! For the money you would not regret it!

Same Issue I use a Tamron 70-200mm 2.8 and a Nikon d300 same thing really and I only get 1/250 It's no Problem just put the exposure to -0.7 will give you about 1/125 more then Put the camera onto Shutter Pri. Set the Shutter Speed to 1/400 Does the Job ISO 6400 Exposure -0.7 - Quality RAW and Jus lighten the Images afterwards also remember use auto Iso but after the first math use one of the Night Images you taken put it In Picture Control and Make a White Balance present to Night shooting , Enjoy !!
Canon 70-200mm would be a perfect choice! Since you said that you had enough budgets, avoid buying second hand.
100-400mm is fantastic......
70-200 on a crop is about the minimum for sideline sports, so on FF you're going to struggle.
Just shoot for the best image quality you can and be prepared to crop. What's the output medium? If you're not expecting to be producing 20" prints, then cropping shouldn't be an issue.
My general advice would have been, the 5d's AF system isn't really up to sports and that FF isn't buying you an advantage either.
But with good technique, you should get some good results.