I am looking for a business/firm which sends dementia-care personnel directly into the home on a weekly, monthly/hourly basis.

Care.com and Next door.com Place an ad .

Care.com

Visiting Angels (a national franchise)

or,

Do a browser search for your local zip code.

We aren't anything but a bunch of strangers from around the world who have dealt with aging relatives and loved ones, or ourselves looking at nearing the end of our lives.
If you are looking to hire help you will have to look in your area.

Do scroll down to the bottom of the page and you will see on the far left some advertising sponsors until the column "in home care". Might be a place to start. I have never used it.