Will Medicare pay for an adult child to care for elderly mother at home?

No. Medicare doesn't pay for any in home help from family. Medicare pays medical bills.

Medicaid programs sometimes pay for home help. Usually not from family however. And usually for minimal care and help.
Check with your state's policies if your senior is without funds and on Medicaid. Look this up online for a start. Speak with elder law attorney or social worker in your area next step.

Do know that here on Forum those we see using the very minimal state help and giving up their jobs to do in home care often end homeless and jobless and without a job history in the end. We have sometimes had to go so far as to recommend accepting homeless shelter to try to get back on their feet after the death of a loved one and "clawback" on Medicaid funds takes everything.

"Michigan's Home Help Services Program

Michigan also operates a program called Home Help Services for seniors who need extra assistance to stay in their own homes. In Home Help Services, you hire your own provider to help you with tasks of daily living that you need assistance with, like bathing, eating, getting around inside your home, light housework and laundry, and some shopping. Medicaid pays the cost of your provider.

To qualify, you must meet Medicaid's income and asset requirements, and your doctor must submit a form saying that you need help in your home. Then an assessor from the Michigan Department of Health & Human Services will visit you and evaluate how much assistance you need with activities around your house. Medicaid will tell you how many hours of assistance it will pay for in a month.
To apply for Home Help Services, contact your local office of the Department of Health & Human Services.

Assisted Living Facilities in Michigan

If you are enrolled in Medicaid or MI Choice (see above), your regular benefits will cover some of your expenses while you live in an assisted living facility, like physical therapy or prescription costs. However, Michigan does not have a Medicaid program that will pay for room and board in an assisted living facility.

Michigan's PACE Program

Michigan operates a Program of All-Inclusive Care for the Elderly (PACE). To qualify for PACE, you must be 55 or older and live in an area served by a PACE provider. You must also require a nursing home level of care, and you must be able to live safely in the community with PACE services. You can see a list of Michigan's 18 PACE centers here.

PACE participants receive their services from an interdisciplinary team of professionals like physicians, nurses, and social workers, whose role is to coordinate individualized care and services to keep seniors in their own homes and communities. Medicaid pays for all the services offered in the PACE program. If you are interested in a PACE program, apply directly to the one you are interested in, and the program will help determine your eligibility.

Supportive Services for Michiganders Who Do Not Qualify for Medicaid

If you don't qualify for Medicaid and do not meet the nursing home level of care, you may still qualify for some personal assistance services if you are physically disabled and need help with activities of daily living like eating, dressing, or bathing. Michigan's Area Agencies on Aging administers a supportive program. Contact your local AAA to apply."

Source: https://www.nolo.com/legal-encyclopedia/medicaid-michigan-residents-needing-assisted-living-home-health-care.html

This may not answer the question about whether you, a family member, can be paid, but it will probably not be the best arrangement since you'll probably only get minimum wage and it won't be for full-time hours (and then all the other hours your provide care to your Mom... you will be 24/7 so think this through very carefully).

Your Mom would probably be better off socially in AL (and you can retain your own life balance). Think about what will happen if/when she is no longer mobile or well or cooperative. Read some of the posts on this forum under the topic Caregiver Burnout. Don't think it can't/won't happen to you. It happens to the best of loving, well-meaning adult children who have good intentions but then don't count the cost or don't have their eyes fully open before taking this path. Regardless of whether your Mom resists AL or not... the caregiving arrangement needs to work for both the receiver and giver.

I wish you much clarity, wisdom and peace in your heart on this journey with your Mom.

I did a search, and here is an article that you may find helpful, Tracy:
https://www.payingforseniorcare.com/paid-caregiver/elderly-parents