It's certainly true that almost all that was published after the first split brain operations in the 60's and 70's was wrong. But there is lots known now about brain lateralization which is on solid scientific grounding. But the way it was talked about back then almost like a way to sort personality types was all half baked and has been debunked.
As you say the two halves collaborate on everything. The idea that some functions were handled on one side while others were done on the other was another mistake. As it turns out every creature's brain is lateralized as ours is and for the same purpose: to attend to two things at once. Broadly one side looks out for dinner and other needed tasks while the other watches out to avoid becoming another creature's dinner. Those tasks require different modes of attention.
For all creatures, manipulating the world to attend to routine needs happens on the left side where focused attention is often required. But survival requires broad, sustained attention in an open ended way to respond to whatever comes up. Deliberative thinking takes place in the left and constant vigilance takes place on the right side. The person who has done the most to assemble the more recent research is Iain McGilchrist and the has put the information out in two ambitious books, The Master and the Emissary in 2009 and The Matter With Things in 2021.
My introduction to his thinking came through this animated video which was created to accompany a talk he'd given. It is blessedly simpler than the books though those are very well written.