The Wheatcroft patent has been discussed here several times, and IIRC was initially brought up by PE.
Although it can be shown theoretically that the conversion does not take place, it can also be verified experimentally: prepare a batch of developer per the patent and a second batch, where the H2O2 is added to the sulfite in absence of HQ and the rest of the developer mixed. Run parallel sensitometric tests. Alternatively, the free HQ can be extracted with ethyl acetate and quantified.
Oxygen can be used and the reaction proceeds as expected, but again, it is pH dependent and produces hydroxide, so either a strong buffer system has to be used or pH correction has to be applied continuously. This still means the liberated Na+ has to be bound up as some salt, the acetate or borate being the least troublesome. I thought I can be clever and use SO2 gas to neutralize the NaOH back to sulfite but it cannot work in a batch process, and it becomes a frustrating juggling act to do it continuously.