Building muscle tissue is fairly straightforward. if your legs are limiting you there are numerous training techniques which Will work if you do them properly and repeatedly over a span of time. Increasing lung capacity is a lot harder, it seems to me, and increasing VO2---basically gas exchange for fueling and exhaust ... there is a reason the pros use EPO.
I am not a doctor nor an exercise specialist or really much of anything ... but as i understand it the legs burn because (I think (carbolic acid (CO2 (cell exhaust) mixing with other compounds) becomes lactic acid and actually is burning your capillaries if you cannot extract it quickly enough. When you have reached your maximum volume of gas exchange, your cells cannot pick up waste mater and deliver fuel as fast as it is needed. The muscles starve and weaken, and the waste builds up and causes pain.
Therefore, better breathing Should increase fuel delivery and more important, waste evacuation, which would lessen the burning sensation.
I am sure a dozen people at least will tell me 20 ways I am wrong, and at least of third of those posters will use facts, and about a fifth will use relevant facts.
I use google. Not only are my odds of finding the real facts a lot higher, I don't have to listen to people say things like, "Studies have shown that lactic acid isn't a thing and wouldn't matter if it was."
I didn't google any of the stuff above, so you have about 40 percent odds of 40 percent of it being accurate---you decide which portions you want to believe. That's how it is done, nowadays. Alternative facts rule!