I have a son and daughter at KGS and we love it.
The difference between the two schools is that KGS is coed, so there are fewer boys than at Hampton which affects the number of teams KGS puts out. And yes the facilities are not on site, but it is not a problem, they go for a sports afternoon to Ditton Field and just go home from there, same with after school training or matches. There are a lot of matches ob Sats if he makes the team.
A new director of sport has recently taken over and this has led to changes.
In the winter boys will do one term of hockey (taken v seriously at KGS) and then one of football. In the summer they will do cricket and some athletics.
Children with aptitude in particular sports are invited to attend team training and so play for the school.
From the Third Form, rowing becomes an option and this starts as one afternoon and a Saturday morning, then increases to after school training. Rowing takes over your life, my daughter is very serious about it, but I like that! If you are a rower, you row all year round, you give up everything else in terms of sport. Rowing at KGS is absolutely excellent and run by very committed coaches. They regularly get kids into the GB training programme. Hampton may do, I don't know. The Boat House holds the most stunning location of alk the schools who row - right opposite Hampton Court. It is being rebuilt this summer and will be even more gorgeous!
In addition to this, boys can do a six week tennis course in 1st, 2nd and 3rd years on Sat mornings.
There is also some cross country running.
There are a lot of House Sport competions - today it was House Ergos which my daughter and her team won 
There are the most enormous number of extra curricular / after school club type activities at KGS including I think fencing, table tennis, CCF (target shooting, outward bound type activities)
In first and second years they do a six week swimming course at the Kingfisher.
They also do PE once a week but that is at school or on the field behind.
The sports taken very seriously are rowing and hockey. Everything else is very much lesser.
The new director of sport is a rugby man, so I wonder how long it will be before rugby comes in? Maybe never, who knows.
Hope this all helps!