What is a "Tetrode"? Is it a Pentode, with one grid not connected?
Anyway, in the "Ultralinear" design, the "extra" output tube grid is excited directly with the signal from a winding of the output transformer made for this purpose, and provides the simplest and most effective form of negative feedback around the output transformer. Output transformers are highly nonlinear and do not have flat frequency response, which is why you can't get much poiwer through them in triode mode. The Pentode connection lets you drive more power through the transformer before distortion becomes excessive. That's why Pentodes were invented.
Anyway, in the "Ultralinear" design, the "extra" output tube grid is excited directly with the signal from a winding of the output transformer made for this purpose, and provides the simplest and most effective form of negative feedback around the output transformer. Output transformers are highly nonlinear and do not have flat frequency response, which is why you can't get much poiwer through them in triode mode. The Pentode connection lets you drive more power through the transformer before distortion becomes excessive. That's why Pentodes were invented.