Yes, you can, but what one has to understand is the heat diffusion in a grain bed. Left static will take hours to equalise. Stirring the grain bed constantly will get all the grain to the set mash temperature, and stirring it constantly will keep it there.
It would be ideal if we could dough in and the thermal energy equalised immediately throughout the grain, unfortunately, it doesn't work like that. Doughing in higher to allow for the grain temperature is all well and good, as long as one is doughing in slowly while keeping a close eye on the temperature. Mistakes made in dough in, and mashing temperatures lead to denatured Beta Amylase and a lower fermentability of the wort.
Stirring will also prevent a stuck mash, if using an adjunct like flaked corn, oats, and barley I stir constantly for 20 to 30 minutes until the starch has broken down.
So what does the pump and recirculation of the wort do? Nowhere near as much as stirring as far as the heat diffusion process of the mash is concerned, but as its name suggests recirculate the brewing liquor.