I typically mash in converted cooler MLTs because I often make batches too large to fit the full volume in my kettle, though I absolutely see the value in BIAB and find myself utilizing the method on smaller batches with absolutely no ill-effects.
When using a fabric filter and the full volume of mash water without sparging, the gravity is set when the mash is complete - unless adjusting for volume in the kettle or the fermenter - it is never diluted. In addition the pH is fixed in both the mash tun and the kettle. The entire sparge process can be eliminated, saving time, effort, equipment and energy expense. The use of one kettle, one burner, and one bag with no mash out produces comparable results to sparging.
Squeezing the bag of grain and tannins are not synonymous. Excerpted and paraphrased from the books “Water - A Comprehensive Guide for Brewers” and “Malt - A Practical Guide From Field to Brewhouse” - Tannins, a subset of polyphenols, are present in grain husks and cell walls. They are released at mash temps and bind with proteins to form haze. In conjunction with a pH above 6, excess tannins are extracted and impart an astringent flavor - they can not be produced by pressure.