I'm sharing my personal AOS experience of trying this approach (without a lawyer). If you see my timeline, I'm in the very early stages of the AOS process. I'll provide future updates of my AOS progress.
As you can read for yourself in this recent cheat sheet for immigration lawyers, two different circuit courts have found that when USCIS grants TPS it counts as an “admission” for purposes of AOS. The two cases are Flores v. USCIS, 718 F.3d 548 (6th Cir. 2013) and Ramirez v. Brown, 852 F.3d 954 (9th Cir. 2017). But you can only use this approach if you reside in the Sixth Circuit (Kentucky, Michigan, Ohio, and Tennessee) or the Ninth Circuit (Alaska, Arizona, California, Hawaii, Idaho, Montana, Nevada, Oregon, and Washington (state)). Regular AOS applicants usually submit a copy of their I-94 as proof of admission, but I provided a copy of the 20 page Ramirez v. Brown decision (because I reside in the Ninth Circuit).
I concurrently filed the I-130 (by my USC 21 year old son), AOS, EAD, and AP. I'll post as replies below my cover letters for each application (without personal info). Feel free to ask any questions if you are in the same situation.