The above without taking your eyes off of your target .
I do mine a little defferent when shooting match rifle.(rifle never comes off the shoulder nor cheek off the stock)
When shot breaks index finger comes out of trigger guard and up side of stock lifting bolt and slide over the bolt, little finger hooked to catch bolt and bring to rear. Then push hand forward, heal of hand catches bolt moving it closed and locked and right back to grip.
This is done with a rolling motion back and forward. It give me time to check my spoting scope for wind change while reloading from the striper clip and at the end of the string before the targets go down.
In sitting elbow never leaves my anchor point of my leg , in prone it never leaves my mat all movement is with the fore arm and wrist.
Shots made standing while hunting is done the same.
You are, of course, 100% correct. But like most things "gunny", most people haven't been properly trained.
When using a firearm, doing what seems "natural" is often completely wrong. That is why training is so important.
Not meaning to hijack your thread, but the British trained thair soldiers to use a very simular system to operate the SMLE .303 rifle.
They were taught to operate the bolt as you suggest and then use the right middle finger to pull the trigger and then quickly operate the bolt again. The result was very rapid fire. The Germans, when first confronted by this system, thought that the Limeys had developed an automatic rifle.
But it was only the SMLEs, operated smartly.
Today's history lesson, by Old_Painless.