This is a Delta requirement. If you read the actual Delta regs, they state that it must be FAA approved in order to be brought down the jetway.
It is entirely up to the gate agent whether it will be valet checked or through checked. I was an a MCO-MSP flight two weeks ago that had gate checked items sent to the baggage carousel rather than valet checked to the gateway.
Here is the actual Delta page about car seats and strollers.
https://www.delta.com/us/en/baggage/special-items/children-infant-items
To answer the OP's question: You would need to wait in line to have it checked. Depending on where you're flying from it might be the same line.