If a loan officer takes a verbal application, can they just document the information on the application form and then mail it to the application for a signature?

Assuming the loan officer gets everything needed for a complete application, the 30 day decision timing starts with the date of that verbal application and would be tracked from that time.

Does this sound correct? Anything I should be concerned with?

Thanks

There's no requirement for a signed application, especially if it's taken by telephone.
Assuming your LO is going through the 1003 over the phone with the borrowers, you have an application for RESPA as soon as he gets the six items.
Your 30 decision clock doesn't start until you have sufficient info/documentation to make a credit decision.
Normally that happens after you get the intent to proceed.
Around here, the GFE/TIL go out with a letter saying 'if you wish to proceed, we will need the following:....'

Actually, as soon as you have an application you have 30 days to approve (if possible) Or deny on what you have or issue a notice of incomplete application. Once you have a complete application, the clock starts and you then have 30 days.

From B:

"(c) Incomplete applications. (1) Notice alternatives. Within 30 days after receiving an application that is incomplete regarding matters that an applicant can complete, the creditor shall notify the applicant either:

(i) Of action taken, in accordance with paragraph (a) of this section; or

(ii) Of the incompleteness, in accordance with paragraph (c)(2) of this section."

If you are waiting for things outside of control of the applicant, the clock is not running but you are expected to try to obtain the quikly.