First, i said lactate, not lactic acid. Yes, it's pedantic terminologies, but there's an important distinction

2nd, key word there is once exercise is over. This would be the case for bouts of anaerobic exertions, where repeatability of intervals depends on initial amount of glycogen available and rate of lactate clerance by the liver before the next bout. If the OP is accumulating lactate 10 minutes into an aerobic exercise, in which lactate shouldn't build up appreciably, it would indicate that the OP's muscles don't have enough oxidative enzymes. Then, the only thing to do is to stop, which would defeat the purpose of aerobic efforts.

It's also possible that what the OP is experiencing is not soreness but actual strain and tearing of muscle fibers as those fibers can't handle the exertion required, in which case, recovery is needed for the muscle to repair itself, and it should get stronger as a result.

edit: that article is full of BS. To read Kolata talking about science would be the same as going on Velonews to read reports on the crisis in Syria.

Lactate is not a fuel, period. It could be converted into fuel in the liver via the Cori cycle in a process called gluconeogenesis, but muscles can't directly metabolize lactate. Lactate concentration probably also doesn't have nearly as much effect on terminating exercise as the build up of protons, but that's another issue; this is why some try bicarbonate regimens so that there's more buffer in the blood.

Dr. Phil Skiba was kind enough to share his research on the rate of gluconeogenesis, and I'd suggest you and the OP read it. The take away is that the rate is painfully slow, and it's nigh impossible at any significant levels of exertion.