You might want to quit using chat bots to answer questions. I’d bet a regular Google/Brave/Duckduckgo/Bing/etc search would give you much better results.
The primary for old people is Medicare, not Medicaid.
Absolutely not. Medicare will not fund nursing homes for elder care. It will fund healthcare, but not day to day life in a nursing home. Medicaid is the source of money for almost 70% of all residents. It is all so very expensive that personal assets are soon exhausted (if there were any to begin with) and at that point Medicaid, not Medicare, steps in.
What was the link you found that the AIs did not find that said Medicare funds elder care nursing homes living expenses.