"I don't know of any private or public major medical insurance plans that cover nursing home care, long-term care, or custodial care."
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Literally a 5-second Google for supplemental Medicare and health insurance will pull countless providers who cover this service.  Whoops.
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"So it's really not surprising that Medicare does not cover these services either."  -- Yeah, so all those elderly you say "love" Medicare? Yeah-- they don't "Love" it if they need all the services listed above.  And you're correct-- what Medicare and private insurers are willing to cover only decreases by the day.