Two separate corrections, and it is worth keeping them apart.
First, lactic acid as such barely exists at body pH. It dissociates almost completely, so what accumulates is lactate plus hydrogen ions. The acidity comes from the hydrogen ions, and lactate is produced alongside rather than being the acid.
Second, lactate is not waste. It is shuttled to other muscle fibres, the heart and the liver and used as fuel, which is the Cori cycle.
The burning sensation tracks hydrogen ion accumulation and the metabolites that build up in hard anaerobic work, sensed by metabolically sensitive afferents. Lactate is the marker that rises alongside, which is how it got the blame.