I must say I am a Match Secretary in an amateur League and can tell you most clubs like to see league schedules punctuated with Cup ties.
In SESL if the league is going well you have the decision to "throw" a cup tie and save ELs or if your league programme is not doing so great it is a break and you might catch teams in the former situation. It is a chance for smaller teams to do something - if not win the Cup then at least make some cash by getting through a couple of rounds.
Leaving all of a Cup competition till the beginning or the end usually results in the highest rated SL teams winning - not an incentive to smaller clubs at all.
However at the moment we have a League Cup section type of Cup - a home and away double leg Cup, so why not a one off KO Cup for each Division, played on neutral ground.

Gives every team a real chance of winning a Cup.
Only a thought.....!