Trad outdoor climbing actually depends a lot less on finger strength than on body positioning and balance. Indoor climbing tends to be face climbing, in which hand/finger strength is important, although the better you get the LESS you actually use your arms, as you learn how to use your whole body (for example, flagging a foot to one side to counterbalance your arm on the other makes it easier to hold on without hand/arm strength).

On the other hand, technique-y climbing does depend a lot on your knees, as you need to be able to (for example) place a foot high and then shift your weight over it and straighten your leg to stand.

Being fat will make climbing more work for you. I have climbing friends who are not sylph-thin, though, who are plenty decent climbers.