There are some terminology problems with this post. "Lautering" is separating the wort from the grain. All wort making processes employ lautering. "Sparging" is rinsing spent grains with fresh water, to rinse more sugar from the grits than you would recover without sparging.
Recirculation is not a substitute for sparging. You cannot get more sugar off of the grain grits by "washing" with wort of the same concentration that you just lautered from the mash. That would be equivalent to rinsing the soapy water off of dishes by dunking the dishes back into the soapy water. Recirculation can speed up the conversion process in some cases by speeding up the gelatinization process, which must (partially) occur before any enzyme catalyzed hydrolysis of the starch can occur. Hydrolysis is the process by which starch bonds are broken to convert high molecular weight starch (non-soluble), into lower molecular weight starch (soluble), dextrins (soluble), and fermentable sugars.
Everything that is soluble in the wort contributes to SG, but only the simpler sugars are fermentable. Fermentability is determined by the weight ratio of fermentable sugars, to soluble starch and dextrins. The primary determinant of dextrin content of a mash that has 100% gelatinization and solubilization of starch, where enzymatic action has been exhausted, and there is only sugar and limit dextrins remaining (no remaining soluble starch or higher molecular weight dextrin) is how much limit dextrinase action occurred during the mash. Limit dextrinase is the only enzyme occurring in malted barley that can hydrolyze the branch bonds in amylopectin, and thus reduce the amount of limit dextrins in wort. Limit dextrinase starts denaturing at slightly lower temperatures than does beta amylase, and is actually the enzyme that can create higher fermentability with lower temperature rests, which have long been referred to as "beta" rests - they are actually "limit dextrinase" rests. Beta amylase is not required to create fermentable sugar, as alpha amylase can hydrolyze starch to nothing but fermentable sugars and limit dextrins. Beta amylase just speeds up hydrolysis and increases the ratio of maltose to other fermentable sugar in the wort, compared to alpha amylase alone.
If alpha amylase alone could not create fermentable sugar, then adding alpha amylase to a low fermentability wort in the fermenter would not be able to provide a reduced FG. A great deal of empirical evidence shows that alpha amylase in the fermenter can increase the fermentability of low fermentability worts.
Brew on