Checked baggage through TSA carry-on security when late for check-in?

Hi Travel Stack Exchange,

Recently on a flight from LAX-BNA, I missed the cutoff for checked baggages by 6 minutes (10:15am departure, tried to check bags at 9:36). Yet, the automated check-in system still allowed me to pay 30$ to check-in my bags but wouldn't let me drop it off. Naturally, the check-in agent refused to accept my bags and told me that I had to "find someone to deliver it for you". (right, like i'm going to do that with 10 minutes before the gate to my flight closes smh).

Anyway, I asked him if I can gate check my bags and he said that TSA won't let me. I also asked him if they could put my bags on a later flight (this happens all the time) and he said that the bags must go with the passenger. Instead, I ended up forking $83.70 for him to book me on a later flight.

Question: Does TSA care about the size of your bags when going through security if people can gate-check their bags "complementary of <insert airline>"

In hindsight, I should've tried to go through TSA with my checked bag and gate check it so I wouldn't have dealt with the bs of forking more money. Ironically for the rebooked flight, both my checked-in bag and carry-on baggage did not make it onto my flight, and as of the time of this writing, I'm waiting for BNA to deliver my bags to my door.

Note: this question exists but does not answer the question I'm seeking.