Why don't baseball teams use more pitchers for fewer innings?

The obvious objection might be that the top pitchers in a rotation would be taken out before they have to, but if this also meant they were available on less rest, it seems to me the total number of innings they pitch in a season could stay the same. Wouldn't it give every pitcher the chance to pitch like a closer (high degree of focus and energy for an inning or two), and also provide more pinch hitting opportunities in leagues with no DH?