I rewrap. In my experience the cheaper the wrap the less adhesive is used. Even wall mart 9 dollar wrap is too sticky for rewrap. The China 3 dollar wrap is easy to rewrap.
I got a used 10 year old 100 mile cannondale on CL to use as a donor parts bike. And the bar tape had a rubbery feel to it, so I put that on as well. I really like it but recently it started to peel. So finally using a BF tip I decided to use inner tube as bar tape. I thought it would be similar to the old tape. It works however it doesn't have as much padding. Thinking of just wrapping over it again with some more inner tube.
Fwiw, I'm not cheap or frugal, I'm just poor when it comes to bike money.