Filling the tires as you go ensures more thorough fill, especially inside the sidewalls. That's pretty important if you're stacking them more than 5 or so tires tall, as the weight from above is going to crush the lower tires. I used a shovel for all of it and it was a painful project. Renting or borrowing a front end loader would help dump the dirt in the right place faster, but you'd still want to hand-shovel/spread the dirt around the tires to ensure you're filling the sidewalls. If you just stacked 3 or 4 tires at a time and then dumped in dirt, you'd end up with an unknown quantity of dirt in each column; when you shoot, you don't know if the bullet will impact several feet of dirt or just some tread and then empty spaces. Most pistol rounds will sail through 4-5 layers of tread.
A trick I learned on the second backstop I built (same use of tires) was that you can fill the bottom row, drive some old lumber or fairly straight branches (stripped of any smaller branches/leaves) into the center of the pile, then continue stacking tires. The effort you put into driving that wood into the bottom tire (sts) takes up some of the dirt you would have had to shovel in, and helps keep the follow-on tires a guide for centering them up. If you have a 4x4 in the center, it will have about 1ft of dirt in front of it. It's going to take a lot of rifle fire before that lumber is destroyed, and even then the dirt is there to settle into the gaps.
After a couple months and a few thousand rounds, I found that the dirt settled quite a bit. I just topped off the top row and let it continue settling. I have yet to have any columns fall over or collapse, though the bottom tires do need to be slightly thicker tread to bear the weight.
If I had a front end loader, some heavy wooden beams as a support structure, and a supply of relatively rock-free dirt, I'd go for the straight dirt berm. I have none of those, and the tires cost me $0 from some patriotic folks in town. When I go to sell this place, I'm banking that the potential buyers are as enthusiastic about a large pile of tires on the land as I am. Otherwise, it's going to be a PITA to tear it all down.