I would argue you are actually too big to be playing in that competition. You have 3 Saturday teams playing a good standard, whether technically you are a village or not it's going to be hard for a traditional village team. If you are a 1 team village side (in the spirit of maybe the original competition) then why would you want to play against a good setup like yours and get smashed? It's not your fault but maybe to the grow the competition it needs to split up the bigger more professional village setups from the typically english village setup. My club, Ashorne, has 1 Saturday team and no way would be want to go into the competition and face some of the teams in our area.
I see your point and we've seen it at a different level in the final a few years ago. They had two lads that played under 17s Lancs, one guy that played Lancs 2nd XI and an overseas that had played against India a few years before. They were all within the rules set by the comp as none of the matches were first class but against my DJ, Delivery driver, Carpenter, Farmer etc it was apparent, as clubs, we had completely different set ups and there was a major gulf between the two.
I don't know what the original rules where but I do know the organisers have had a problem with attendance for a few years so restricting to a one Saturday team team set up, to play on a Sunday, could risk lowering the number of teams further. Equally it might increase, if a few teams knew that wouldn't come across anyone too strong. Who knows! At the moment, it's a good comp and has a lot of teams taking part that wouldn't have an opportunity or stand a chance in a nationals, I guess it's where you draw the line that is difficult. I think if your team is strong enough to play in a county league (which we do) then you probably shouldn't be in this competition.