I don't have much use for shoot-thru mounts, as most of my rifles either came without sights, or I took them off. I do have shoot-thru mounts on my pump .30-06 carbine for jump-shooting deer on cut-overs. It's rare to have a standing shot when walking through cut-over trying to jump bedded deer.
As to shooting moving game with a scope, I killed a doe two years ago with that same .30-06 while she was running at about 125 yards. I didn't even consider using the open sights. I have found that with a little practice, finding your target in the scope comes as second nature. You just have to make sure your scope is always turned down. Distant shots usually allow time to crank the scope up, but close shots rarely allow time to crank it down. I had a brother-in-law (now deceased) who used to pratice with his .270, scoping birds as they flew by, snapping and bolting, snapping and bolting....it was awesome how fast he could shoot that rifle at running deer. He once jumped three deer together, which proceeded to go in three different directions. He got all three.... Armadillo hunting is a great way to hone your "moving target" skills. We drive around in the fields at night shooting them. Often they take off running so you have to get them on the go. It's a hoot.
If you feel it's difficult to quickly acquire your target in your scope, you can practice (literally anywhere) on improving that. Simply check, then double-check, that your firearm is unloaded. Pick a spot on the wall, a knob on a cabinet, an insulator on a power pole....you get the picture....now as quickly as possible, raise your gun and scope whatever it was. Lower and repeat. Practice at varying scope power settings. I can usually find my target rather quickkly, even at higher power settings. I do this quite often though. When you get to where you can acquire a cabinet knob ten feet away without even thinking about it, a possom/coon/fox at thirty yards is a piece of cake.
