Do TSA regulations allow CFIs to start training non-citizens before the TSA has approved their training request? 49 CFR 1552.3 says (emphasis mine):

(4) TSA has informed the flight school that the candidate does not
pose a threat to aviation or national security, or more than 30 days
have elapsed since TSA received all of the information specified in
paragraph (a)(2) of this section.

This one implies we can't train until we hear back from the TSA for aircraft that are in &quot;Category 1—Regular processing for flight training on aircraft more than 12,500 pounds&quot;.
However, the section on &quot;Category 3—Flight training on aircraft 12,500 pounds or less&quot; says:

(4) The flight school begins the candidate's flight training within
180 days of the date the candidate submitted the information required
under paragraph (a)(2) of this section to TSA.

So that tells me that as long as the information is submitted, we can begin training a non citizen immediately (for less 180 days).  What is the point of this?  Where is the security?  Is this correct? Or is it just another confusing regulation?
What is the current industry practice? Do CFIs provide the training in this situation? Or are more TSA checks initiated?
This is addressed in the AFSP Candidate FAQ, which confirms that you can start training in category 3 as soon as all information has been submitted:

How long should I expect to wait after I have submitted my training event request?
[...]
Candidates in Initial, Instrument, and Multi-Engine training (category
3) may begin training as soon as AFSP receives all required
information, including the candidate's passport, fingerprints, the
fee. You and the provider will be notified once AFSP receives the
fingerprints and you have been granted Permission to Initiate
Training.

That's also what the regulation says, but the FAQ may be a little clearer.