More or less. The direct O2 gain is rather minimal, and as Eric wrote, the loss due to the muscular effort may be in fact bigger.
Larger, deeper breaths also raise HR, at least in my case. I use a couple purge breaths and deep breaths after surfacing (after hook) as part of recovery when spearfishing or doing challenging pool sets, but prior to actually dropping breathe very minimally. My best dives/holds always begin with my HR low and I think myself to as relaxed a state as possible...
I think the 'breathing' part of breatheup is a myth but not the relaxing part... Although starting from a rested state the relaxing part only takes seconds.