There is ZERO chance of a closer ever pitching in a non-save situation to "get work in" under your scenario. Combine that with no back to back days and I'm guessing your average closer is going to be down to about 50 IP. No chance they pitch as many innings as they do now.
The only way top bullpen guys pitch as many innings is if they are more frequently used in multi-inning appearances, which is probably going to lead to them giving up more runs.