Technically, yes -- this is a comma splice between two independent clauses, as you say. However, this punctuation rule is relatively recent in English, and, if I had to guess, didn't become codified until sometime in the 19th century. Prior to that, a comma designated a brief pause in the flow of the sentence, as if to take breath. Where does your text come from? I tried googling for it and couldn't find it.
Even now, some authors use it, and I can see why they might in your example. The problem in fixing it is that none of the approved alternatives are ideal, because you have three thoughts of more or less equal weight, and that should be reflected in the punctuation. You could use a conjunction -- "and" -- to join the first two clauses, but that gets repetitive with the next "and", and it's not what the chap said anyway. If you replace the first comma with a period, you satisfy the rules, but you have separated the first clause from the other two, and the thing becomes unbalanced. A semicolon would be less disruptive and is "allowed", but the second comma would still make the clauses unbalanced. The only thing you can do then is to replace the second comma with another semicolon. If the clauses were longer, and especially if they contained internal commas, that would be a fine thing to do, but with these short clauses, the whole thing becomes rather disjointed. So commas really work the best here after all, and reflect how you would actually say the sentence aloud.