There seems to be a lot of confusion, so here is my big picture understanding.
There are 2 big things going on regarding aerobic performance.
1) Vo2max, or the maximum volume of oxygen that the body can consume during intense, whole-body exercise, while breathing air at sea level. Its more or less how much oxygen your heart can deliver to your body. We know that as a whole, the capacity of the body's skeletal muscles to use oxygen is greater than the body's ability to deliver oxygen to its skeletal muscles, so a limiting factor in aerobic ('with oxygen") exercise is how much oxygen the body can deliver to the muscles.
Recent studies focusing on the genetics of exercise adaptation have also demonstrated that not only is our starting point genetically determined, but our adaptability to training (how much we improve) is also quite variable and genetically influenced. While the typical person will show a substantial increasing in VO2 max with 6 months of exercise, carefully controlled research studies have shown that a small percentage of people will hardly show an increase in VO2 max at all.
Good linky
2) Lactate threshold: Is the intensity lactic acid accumulates in the blood stream. It is a measure of how long you can sustain performance at a given level of VO2 max performance. If the rate of uptake, or disappearance, of lactate equals the rate of production, or appearance, in the blood, then blood lactate concentration stays constant (or nearly so). But, when the rate of lactate production exceeds the rate of uptake, lactic acid accumulates in the blood volume, then we see the ONSET of BLOOD LACTATE ACCUMULATION (OBLA). This is the traditional "Lactate Threshold" (LT), and once you pass you find fatigue on its way. Its sounds like the OP knows about this, as evidenced by his comment that:
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It looks like my heart and lungs are essentially OK, and anaerobic work is fine, but aerobic exercise doesn't happen. It matches how I feel when I run: within 2 minutes my legs get heavy and it becomes very uncomfortable (in a lactic acid way that feels more like high reps lifting) well before I get out of breath.
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I focused on your title at first but this comment is insightful as to your situation.
A big VO2 max sets the ceiling for our sustainable work rate. It is a measure of the size of our performance engine. However, the Lactate Threshold greatly influences the actual percentage of that engine power that can be used continuously.
Training results in a decrease in lactate production at any given exercise intensity. Untrained individuals usually reach the LT at about 60% of VO2 max. With training, LT can increase from 60% to above 70% or even higher. Elite endurance athletes and top masters athletes typically have LTs at or above 80% of VO2 max. Values approaching 90% have been reported. The lactate threshold (or thresholds) is/are both responsive to training and influenced by genetics.
good linky
Steady state is better for lactate threshold. In most individuals, lactate threshold is the limiting factor in endurance activities. It is also the one that is most susceptible to adaptation from exercise and is more genetically susceptible to manipulation through exercise than VO2 max.
Alex-db, while increasing your VO2 max as your body will allow will be a boost to your performance, it is your lactate threshold that should be your focus, both because it seems to be the limiting factor in your performance and it is more susceptible to manipulation through exercise.