OK i'll be arsed
Clubs can opt to play either day. There are more teams in the SAASL which play on Sundays, and on the whole the Sunday league is the stronger of the two. Not to say that Saturday's league doesn't have decent teams or players, but the overall depth and quality of players is unquestionable.
This is largely due to Sundays having been established far longer than the Saturday league, being created as a workmens league back in the 60's( maybe late 50's not sure). The players/workers usually worked overtime on Saturdays and had only Sundays free. Our Feds also play Saturdays, like you guys, which obviously takes away potential support and some playing stocks from the saturday am's.
You may have noticed we have a large contingent of mummy's boys and religious freaks in SA. These are the people that play on Saturdays, as to not incur the wraith of their mother or their God. (At Easter time or Mother's Day)
A lot of the clubs in the top 2 Sunday divs have grown well and are well supported and organised, again more than Saturday, and recent grants have helped improve a lot of clubrooms and facilities around the league and for any journeymen, is the day which gives them a better chance of earning some decent coin.
Some clubs have switched days in the past for various reasons. Maybe due to an influx of altarboys or tradesman, again not sure.
The only way I'm aware players can play both days is in the rare event of a club having fixtures on both Saturday or Sunday, as they are only allowed to be registered to one club. My club being one of these rarities, off the top of my head, can't think of another apart from Parafield Gardens.
The cup each year comprises both leagues and the drawn home team dictates which day the fixture is played on, so on cup rounds there are numerous clubs playing on their non preferred day. This year's A & B finals were both Sat v Sun with the Sunday clubs prevailing.
Hope that answers at least some of your questions.