For pianists, the idea of always and forever having to work a piece by separately practicing the right and left hands is absurd. What excellent pianist have you ever seen sight-reading first the right and then the left hand? The skilled player uses both hands at once. Separating the hands is a function of your overall piano abilities – as a beginner you may separate many or most pieces, as an intermediate you’ll separate some pieces, and as an advanced player (and always) you may need to separate sections of a piece. However think of each hand like gluing two pieces of wood together – you put glue on both sides and let glue dry to “tacky” and then clamp the wood together. If you let the glue dry completely, the pieces won’t join. So don’t perfect long sections of a piece in the right hand and then the left hand; the hands will become almost unrelated to one another. The piano is a – duh – two handed instrument. Rather, when each hand has moderate skill glue them together while they still need each other. And use speed (slow it down) as another way to make passages come within your abilities.
Allowing slight mistakes of time & note while you perform pieces (in practice) gives enough wiggle room to bring a song into your abilities.