Is there a way the VA could actually help with her brother, rather than add to her current difficult personal situation?

In defense of the Sibling. My Father had his second stroke and was in a rehab facility out of state. They contacted us children and said you have to come and take him. He cannot live alone anymore. My dad did come to WI and enter King Veterans Home only after I had him admitted to the local VA hospital because he called an ambulance for himself stating that he thought he was having another stroke. He was unmanageable and had been drinking. After two months in the Psych ward the Social Worker got him into King.
He later recovered enough to leave King and go back to Georgia where he lived alone even though he had Dementia. My Dad applied several times for Aid and Attendance while living on his own and was denied. He continued to live alone until he fell and laid in his apartment for who knows how long before he was found. After that trip to the hospital he went to a nursing home. I was very upset when I saw the condition of his apartment. He had severe Dementia and was unable to care for his place or finances or himself.
A young couple had befriended him and emptied his bank account every month of his social security check for over a year. He was surrounded by filth and dog mess. Since I was out of state it was hard to keep track of what was going on. The VA was helpful when he was here in WI but not when in Georgia.
Always encourage people to use the local VA office to advocate for your loved one because the National office tends to be a giant cavern of lost paperwork. Copy all applications and paperwork sent to them in case it gets lost.
Any applications for service that are not acknowledged in 6-12 months should be followed up on. You may need to reapply if you have not had an answer
in 12 months or more.
Remember the squeaky wheel gets the grease!!
It is helpful if your loved one is receiving VA medical benefits at the local VA hospital when you apply for services. Engage the Social Worker at the facility to assist when any applications.
Good Luck!

Forgot to add that the VA does have facilities for veterans; your friend and her brother need to get in touch with his primary care doctor at the VA to make arrangements for him to get into one of these facilities.

I find it hard to believe the VA told this individual there was no other place for him to go. I think that might be your friend's perception, and perhaps she misunderstood, or perhaps the brother was confused.

The VA in fact has 3 nursing homes in South Carolina:

If you can manage it, tour them for your friend and help her decide which one to accept, assuming one is close enough for her to visit.

The VA also has a program akin to aging in place. I don't recall the name or details, but it may be that the friend and her brother opted in for this program and the friend is now finding caregiving overwhelming.

frqflyer is right, she agreed to take him and now she realizes she is in over her head. She needs to call the VA and ask them to find a place for him. They can do this, but she has to be willing to give him up.

Somewhere along the way the elderly friend had to have said yes to taking care of her brother. Maybe she thought she could? What are the brother's medical issues that he needs to be waited on hand and foot? Or is she and he of the generation where they think it is the female who does that type of work?

On the Aging Care website are excellent articles. I found one section that deals with Veterans assistance.
https://www.agingcare.com/topics/138/veterans-benefits
Another suggestion, besides the VA, not sure how this works, where he could qualify for Medicaid, thus be placed into a continuing care facility if he is unable to care for himself.