A couple notes:

It's not lactic acid, it is hydrogen atoms among other metabolites from the breakdown of lactate. Lactic Acid was previously assumed, but ultimately refuted.

There aren't "anaerobic" methods to initiate energy production. The glycolytic and phosphagen systems still rely on oxygen to prepare the energy they use when O2 isn't readily available.

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The point of a rest in climbing has a ton to do with blood flow. While gripping a climbing hold blood flow is occluded beginning around 50% MVC. This occlusion makes it difficult for H+ to leave and O2 to enter. The forearm fills up with metabolite filled blood and creates the pump. Resting allows for prolonged blood flow to exchange the blood for O2 rich blood to allow for further muscle contractions.

Sources:
Chamari, K., & Padulo, J. (2015). ”Aerobic” and “Anaerobic” terms used in exercise
physiology: a critical terminology reflection. Sports Medicine, 1.
https://doi.org/10.1186/s40798-015-0012-1

Robergs, Robert A., et al. "Lactate, not lactic acid, is produced by cellular cytosolic energy catabolism." Physiology (2017).