OK, here is what would have had to happen if she had a claim filed that was then sent to DDS.
IF she didn't fill out a paper 3368, then they take that info AT the interview. Verbally. IN fact, there is no need for an in person interview at all if she had filled out the 3368 and mailed it in. I have none of my people ever participate in an in person interview, because generally (not always) they can be damaging to the claimant. So we do he application online, which includes the data that would otherwise be in a 3368. But if, for instance, the online system will not accept the individuals purported credentials and can't seem to match her to its system (and so rejects the online filing possibility), then paper form it is.
Now, those that broach the process without representation and start the process in person, yes, that interview is where they collect the 3368 info, the most relevent of which is the medical provider contact info, so the SSA can order the records at the SSA's expense, rather than the claimant having to pay for it if he or his lawyer has to order it initially.
The 3369, 3373 and 3380 are ALL documents ONLY sent by the DDS - because they are part of the work up, the investigation, AFTER a proper filing of the application has taken place at the local office. The local office, once it has processed the application and the mandatory minimum docs or verbal answers are in (application, disability report data, 827) sends the matter UP to the DDS, who then does its work up/case development.
So, to answer your concern, the 3368 is taken verbally when an in person interview (which can be phone interview or physically at the local office). Sounds like she had a very diligent adjudicator, who should be commended. Most are not that good but occasionally we get a diligent, competent adjudicator. (I am not sure why you have some words in parenthesis.)
Nothing seems suspicious at all. Clerical errors ABOUND in the SSA, particularly the local office. They don't respond to correspondence, they "forget" to process atty rep forms, they even forget to process applications and move them up to DDS. It has been egregious lately. The SSA holds onto money owned by claimant or attorney for months or even years, without budging, without explanation, and with a steadfast refusal to address - getting a live body to actually move on it is like moving heaven and earth. Even my Congressman's team is failing miserably at encouraging this federal agency to do its job (and that is a first, because for the last 10 years, my 2 other Congressperson's always triggered correction and resolution within 2 weeks, 3 at most.
Yes, I prefer to have a written 3368 if one can't file online - that way you KNOW what info you provided the local office because you keep a copy. Paper trail! And anything you give to SSA, bring spare copy to be stamped "received" or "filed" and the date, for your file. Anytime mailed should be mailed certified r.r.r., because the SSA seems to "lose" things with regularity, and the onus will be on your to prove you gave it to the SSA.
I am happy to hear that she finally retained a dedicated attorney. Her medical foundation apparently lacked the strength it needed to establish legal disability, she likely is not learned in the SS laws and what is required for a win, and doesn't know what evidence is acceptable under the law, what makes some evidence carry more weight than other evidence, what statements can be "mis"interpreted to mean something it didn't, etc., and that attorney can work towards strengthening that.