Legs burning has nothing to do with strength. It has to do with training at intensity which helps get rid of the waste products which cause the burning. My understanding is that anaerobic training lifts all limits, aerobic included. That's mostly because it's not really anaerobic, just a small percentage is, but for sure it's at your aerobic limit, which is the point. You know you've gone anaerobic when no matter how hard you try to do deep diaphragm breathing, you start to pant. That's called VT2, or the second ventilation threshold. When you do this, you're not oxygen deprived - anaerobic processes are, by definition, anaerobic. What you have is an excess of CO2 and you're panting to get rid of it.

There are several ways to do this sort of training. 4 X 8' VT2 intervals are very effective, done once a week or as recovery allows. Also over/under intervals are much used. See the 33 workout recipe thread. All high end intervals protocols help.

I'd say that if your legs and lungs are burning at the same time, you're doing the right thing. Lower cadence = more leg stress, less lung stress, and vice versa for higher cadence. So there's that.

IME the answer to your question is yes, but it's more complicated than that. Your muscles would like more oxygen and they'd like to get rid of waste products faster and not only that, but to accumulate them more slowly. On that last point, lots and lots of base are helpful. Increasing one's ability to burn fat helps at all intensities. Burning fat makes virtually no waste products. I try to work the problem at both ends: lots of steady state at VT1 (where one just begins to breathe faster) and as much high end as my bod will allow. I start with zone 4 work, at least 45'/week and when that goes well, add VT2 (zone 5) work. I have lots of hills to practice on, but flats work too.