You want linoleum over carpet which is put on hardwood?
Do it. Follow your Bliss.
I suggest a 5 layer sandwhich of old hardwood followed by shag wall to wall with 16 inch tile squares followed by a thin layer of business tough carpeting topped with a delicate crust of linoleum.
Any GenXers getting involved in Boomer vs Millennial FLOOR WARS should be ashamed.
Only exception is if inspired by Norm on Bob Vila’s This Old House. That is childhood and forgiveable.