Can a home care person just take laundry? It would take about 3 hours.

Our CNA does household chores, including laundry, cleaning the bathroom, washing dishes, vacuuming, etc. in addition to assisting with bathing, dressing and toileting. It's all part of the contract for services with the company we chose, so the extent of the duties will vary by agency.

When you interview home care agencies, be sure to state your specific needs up front. Keep in mind that those needs may change over time, and select an agency that can accommodate those changes as they occur.

The men and women who work as home care aides have been to school to learn basic nursing skills and how to assist people to remain independently in their homes, while many would be willing it is really not part of the job description. What you need doesn't require any skill, just willing hands (and a good personality).

Most places have a wash and fold service. You just drop off and pick up next day. Could you use something like that?

maybe just pay a home aide to come and do laundry at your home?

If all it is that you need help with is the laundry many Laundromats have a drop off service. You drop off and they wash and fold or hang and you are charged by the piece. (some may charge by weight)
You could also get a caregiver from an agency to come in 1 time a week to help with laundry, light housecleaning. Might it also be possible to hire a neighborhood High School student that wants to earn a few extra dollars a week? As you say it would not take much time.