How would I be able to get reimbursed for the travel expenses that I paid for from my own card?

There is a whole class on DTS TRAX that you were required to take prior to receiving a GTCC that explains all of this. You need to go back and retake the GTC 101 class again and it will explain everything. It takes about 20 minutes

Nothing in your Army career is "automatic", the sooner you learn that, the easier life will be.
You have a DTS authorization for this TDY. The authorization tells you what your authorized reimbursable expenses are:
flight/taxi to/from airports
Hotel
standard meals and incidentals (M&IE).
You can look at this for yourself in DTS, you don't have to rely on someone to tell you.

Your voucher will reimburse you for the exact cost of any flight, and hotel, receipts required.
Your voucher will reimburse you for the M&IE amounts allowed on your authorization, regardless of what you actually paid for food.

The amounts that go to your GTC or your checking account on your voucher are entirely up to you, but typically the flight and hotel will default to your GTC, as you are required to use one for those expenses.

The only thing I put on my GTC is my Hotel, Car Rental and Flight. All three of those have to be on the orders. While Per-Diem might be authorized, and put on the GTC I don't bother.

My understanding is while the $300 per-diem in this example could be spent on the GTC, it will only be paid with the submission of the receipts through DTS. Or, the SM will have to pay the balance unpaid.

I understand the full per-diem is paid to the SM anyway to their account of direct deposit, so the payment flows is a matter of semantics. But, if the expense is incurred through the GTC there is now the additional expense process through DTS, or the soldier now has to pay the balance on the GTC themselves.

Net-Net the SM is getting the same amount of money. Keeping the GTC use limited to the flight, hotel, and car rental (presuming all are authorized on the orders) keeps it clean.