My brother in Jesus Christ paperwork, even now there are already 236 active and formally-registered trademarks which have been granted involving "Jesus Christ", 18 if you limit it to "wordmarks." [0]
Also, as per the concurrent(?) edited-in last paragraph, the basic idea remains even if the scope is, er, less-grandiose.
There are a number of longer phrases which include the phrase "Jesus Christ" that are trademarked (either as logos or as phrases), but that is a very different thing.
So, you could start a new church and trademark "Terr's Church of Jesus Christ", as the trademark covers the organization, but you cannot trademark the name "Jesus Christ" itself.
IANAL but I don't think it'd be blocked for that reason: "Right of publicity" is a per-state affair, where it exists it may not be posthumous, where it is posthumous it may not include historical figures, and even if it could apply, there's there's no credible estate/heir of Jesus Christ with standing.
... But I gotta admit it would be amusing to watch someone try to fill that role.
As an example, see Pirone v. MacMillan [0], where there was no posthumous "right of publicity" for Babe Ruth's daughter to draw upon. (At least, not back then.)
[0] https://law.justia.com/cases/federal/appellate-courts/F2/894...
Notably, from your link:
> Therefore, a trademark is "not property in the ordinary sense," but only a word or symbol indicating the origin or source of a product.
> Although its registration is limited to the words "Babe Ruth," Pirone would have us read her rights in that word mark to include every photograph of Ruth ever taken. We decline to do so.
The product in question didn't use Babe Ruth's name in any way implying an endorsement or origination of the calendar; it merely included some pictures. The ruling determined that trademark does not apply here.
Pictures have been deemed eligible for copyright, but that's neither here nor there. The name "Jesus Christ" is not eligible for trademark.
That obviously can't apply to use of the name or imagery of Jesus Christ; there is a much more important federal rule that says it is everyone's God-given right to assert directly that they are endorsed by Jesus Christ.
(This comment is imbued with the divine grace of the Messiah.)
1. It isn't possible to trademark the name "Jesus Christ";
because
2. This might imply that your company is endorsed by Jesus Christ.
But that is clearly false. Those grounds are illegitimate; the government cannot take a position on whether you are or aren't endorsed by Jesus Christ.
There are no "false endorsement" concerns raised by the hypothetical trademark.