You implied that you don't need to own an affilliated resort to join RCI. I think that you are incorrect. On the RCI website, it says: "RCI’s core business is timeshare exchange – ...To enjoy the benefits of exchange holidays you must own timeshare."
In ii, Marriott owners have a preference period of 22 days. For some locations and lower demand times of the year, non-Marriott owners will see some Marriott inventory. But the inventory is more likely to be smaller units. For high demand locations like Hawaii, Phoenix or Palm Desert in February or March, etc. it would be unusual for inventory to sit out there so long that it still exists at the end of the preference period.
For Hyatt, I don't believe that there a preference period. You just need to have a good trader.
But, you have framed your objective in a manner that is challenging and contradictory - - you want to trade into a Marriott, you want to have high trading power and you want to pay the minimum cost (you didn't say whether you meant up-front or annually with MF). If you buy a platinum season Marriott it will cost you more but trades better. If you buy a Blue or Bronze (low) season Marriott it will be cheaper (purchase price) but won't trade as well. Many people recommend purchasing a platinum season unit. Even if you pay an extra couple of thousand, you will still pay the same maintenance fee annually and will have a stronger trader.
My suggestion is to buy a Marriott. Check out Dioxide's ROFR metric on the Marriott subforum. It'll show you how much people are paying for a Marriott. You can pick up a Grand Chateau (Vegas) 1 BR that will pass ROFR for perhaps $1K or less. If you buy an EOY, it will even cost less initially and annually. Personally, I wouldn't -- but it meets your criteria. One thing you have to ask yourself is whether you want to take one vacation EOY or once a year or twice a year, etc.
I periodically see Sheraton's for free on the Bargain forum. I suspect that they trade Okay. I suspect that you would still have the Marriott preference issue.
I own two 2-BR White/Gold season Marriotts in Palm Desert. I paid about $2K or less each (including closing). I may have been smarter buying a Red/platinum season and paying perhaps double. I split the units and get 4 weeks of vacation. So that lowers the cost per week. Also, having two units, I can reserve at the 13 month point - - so I reserve Thanksgiving week (TDI = 125), so it has reasonable trading power. I had traded into Hyatt's five or six times, Newport Coast, etc. So it trades okay.