When I took over my team in the Quick and the Dead league I had to clear so much cap space to make things work. Ended up having to cut or trade nearly three-quarters of the team! I signed a few decent players at key positions and filled out the rest with one season contract players, just to fill out the team.
I thought I would end up at the bottom of the league and getting a high pick in the next draft. But, hey, ended up 7-9 with mid-round picks! And my dead cap has decreased to under $10M from almost $300M too!
All this to say, you may not be able to win the championship with a team of "replacements," but you can still compete and have some fun. It all comes down to gameplanning and being creative with your players' positions. It can still work.