How did "as" (in English) branch into many meanings that look unlinked?

The problem is with your assumption that the senses listed in the OED are unrelated. This is often the cases when one reads dictionary definitions but in fact, it's not very difficult to see the connections.

The original meaning seemed to be one of affirmative manner (so) and it is easy to see how it could acquire a temporal meaning since manner describes events. Temporal meanings often translate into causality (e.g. since) and it is also easy to see the connection of manner or causality and similarity. So you can see that the semantic field of 'as' is actually quite tight and it is not difficult to see a connection between sentences such as:

• She's as tall as Joan. (similarity)
• As Marie would say, I don't like this. (comparison)
• He ran as if his life depended on it. (manner)
• They came as the phone started ringing. (temporality)
• We couldn't come as the trains were late.(causality)

If you look at COHA, you'll see most of these uses attested in the 1810s. I couldn't find any of the causal uses but that's not surprising given the types of genres represented.