It is now known that lactate is formed even when there is sufficient oxygen for aerobic respiration - seems to be triggered by an intensity threshold...mostly - the body produces lactate even in a resting state.
When doing textbook Tabata HIIT the body in minutes can deplete muscle glucose to levels that would have supplied several hours of energy at a lower state - it is my understanding this is due to the glucose being used pretty much exclusively for type II fibers and very little is being oxidized (some of what is, being used to rephosphorylate creatine since lipid metabolism is suppressed). The body can mobilize glucose at a faster rate than is possible to use, at any level of activity, which is what makes this dynamic possible. This is why HIIT works best when the rate of activity exceeds ALL aerobic capacity (or as much as humanly possible). Most of what's left is converted to lactate and used throughout the body, a good bit of it recycled via Cori Cycle.
I don't know of any modern studies that demonstrated a solid link between lactate levels or even pH or H ion levels and contraction force. There are several that show buildup of inorganic phosphate from CrP impedes contraction. One study using muscle that lacked creatine kinase demonstrated that it could produce over a hundred contractions with no appreciable loss of force, while unaltered muscle started about 15% "stronger" but dropped to a fraction of its start force within 20 reps.
I could certainly be wrong about some of this, but from reading a lot of what's out there it seems like a fairly accurate layman's translation? The science is constantly updating, and I've changed my very limited (!!) understanding to build a picture of how different levels of glycolysis interact with CrP metabolism. I might be holding the book upside down...