Professional care for my elderly mom. Is this crazy?

You can apply for Medicaid. Hiring an RN would cost u $30 to $40 an hr. An LPN $25. CNA's get $10. If doctor recommends a NH, place her. Give yourself a break.

Here are the prices from the agency I use. If you do the math with what you’re looking for, it may be more expensive than a nursing home. In my area (central NY) NH runs about 10k a month. Not counting the expense of purchasing a handicap accessible house to accommodate it.

Hourly:
RN 80 per visit
LPN 55 per visit 
Home health aid, 28 per hour, 3 hour min
Personal Care aid 25 per hour, 3 hr min
Companion/ homemaker 22 per hr, 3 hr min

Daily based on 24 hour shifts:
HHA 645/day
PCA 575/ day
companion /homemaker 485/day

Why do the doctors think she needs skilled nursing care? Which of the ADLs( Activities of Daily Living) is your mom struggling with? (They are Grooming, Dressing, Feeding, Transferring, Toileting) Is your mom safe to leave alone or is she a fall risk? Does your mom wander?
I think you can safely keep someone home unless they are aggressive, a fall risk, wandering or cannot transfer themselves (get up out of a chair or get onto the toilet) even with help. If your mom can still help out with her ADLs, I think your plan to get help into the house is a good one.
Once she needs more than one person to do her ADLs, especially transferring, I think you really need the multi shift support of a nursing home staff.

You're an amazing woman, BettyBoop!  That said, don't you think that (given your age) it might be wise to heed the doctor's advice?  Unless you can find someone who's willing to care for her in exchange for room and board, you will find a nursing home to be less expensive than all the skilled care you're considering. God bless and comfort you.... listen to these comments!

Have you considered moving both of you into an assisted living facility? If you can find one that will accept your mom, you can continue to live with her in a place that is accessible and have the security of available help should she fall. I think it would be less expensive than full time in home help and the purchase of a new home.