As the comments have noted, make the bed is idiomatic and introducing a new word or phrase for the process is fraught.
I think you are getting hung up on narrow senses not only of make, but of bed as well. Collins lists 31 meanings for bed and no fewer than 56 different meanings for make, so constructing a piece of furniture for sleeping upon is hardly the only sensible interpretation of making a bed.
To make is not only to create or construct, but to prepare or put into a fit condition for use, an extension of the sense that to make X is to cause something to become X. Homemakers make house. Bivouacers make camp. In some regions, you still make meat when you prepare food. When you make a horse you have trained it for work.
The OED traces make a bed in the sense of preparing a bed for future sleep to around 1300. At the time, such preparation would have been laying out mattresses and blankets on the floor or on benches, as few had permanent furniture for the purpose. Some contemporary analogues it gives are Middle Dutch een bedde māken, German ein Bett machen, and earlier the Old French faire un lit, as well as facere lectum in medieval Latin.
So to say make the bed to refer to the orderly rearrangement of bed linens is not only customary, but sensible even within the modern meanings of make and bed.
