Is it permissible to follow the Hadith of the Prophet (ﷺ) — the meaning of which is that flatulence is not to be considered [as breaking ablution] unless it has a sound or a smell — in the case of frequent, uncontrollable flatulence, and is prayer accepted in this manner? If this is not correct, is it obligatory to make up the prayers that were performed in this way?

The emission of wind (passing gas) nullifies ablution, whether it has a sound, an odor, or neither. So, whenever one is certain that wind has been emitted, ablution is nullified. The Prophet's ﷺ saying: "until he hears a sound or finds a smell" means knowledge of the existence of one of the two matters, not a condition of hearing or smelling. Thus, if you are certain that wind has been emitted, your ablution is nullified. If you suffer from incontinence such that you cannot find a time to pray with proper purification, perform ablution after the entry of the prayer time and pray the obligatory prayer and whatever supererogatory prayers you wish until the time expires or another nullifier occurs. As for the prayers you performed without purification due to this interpretation, you should repeat them, and the view of the majority of scholars is that their make-up is obligatory.