"In other words, any/all public companies are free to keep the TA function in-house."
Exactly. And that's the catch in Renee's proposal. The control point for the SEC is the issuer, not the T/A.
Currently, filers are obligated to report o/s basically quarterly......make that 4 times a year. To increase that level of frequency would require supplementing the information already provided or creating a separate facility...filing...to provide it.
Rather than creating a whole new source of information why not simply require that the number of outstanding shares be published as part of the heading of not just the Q's and the K's, but EVERY SEC filing. Or, if that's too much to ask, incorporate it into the " FORM 12b-25 NOTIFICATION OF LATE FILING" filings which would provide updates even in the absence of the Q's and K's themselves.
Of course, as we know, this process will not protect investors that rely on the soapiness of the sponge and the heat generated by the cartoon character whose visage appears on it to make their investment decisions.......and there's nothing wrong with that either.