I only tie on cross ties. If I do not have a header person to help me, I cross tie while I am putting my cart to. My cart is last on, first off.
I agree you should not tie with carriage put to but without a header person, I felt safer as a rookie and learning and with my slowness with having my horse cross tied while I got everything right, my trainer thought this safest for me. I have cross ties and a rubber mat set up in the opening of my training pen.
The halter worked beautifully while we were learning. I would take off the halter with reins gathered and ready in hand, then quickly enter my carriage, and when settled in my seat and whip in hand, would enter my training area for warm up. After my lesson, I would stop in the same place, exit, re halter and cross tie, and then remove my carriage first and set it to the side. Then the harness came off in specific order.