I suppose you installed using admin ? What path did you put into your PATH ? Is tlgmr findable by a user ?
It should be a matter of readable folder by everyone, and then tlmgr being on PATH so it is found in R.
Check what has been put in your PATH.
Otherwise, you can set tinytex.tlmgr.path R option to where the bin folder is. THis should make R tinytex found the executables.
Otherwise, if you are open to other installation methods you can install TinyTeX for Windows using
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Scoop (https://scoop.sh/) which offers global installs too. TinyTeX is available here GitHub - cderv/r-bucket: Personal scoop bucket I use as an R user and software engineer
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Chocolatey (Chocolatey Software | TinyTeX 2022.11) which also allow global installation
I'll try to find more time to check if everything works ok on a clean computer, but you may succeed to install before then.
Hope it helps