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Is it clear that the reduction in bid-ask spread is due to HFT rather than some other change in market dynamics? I frequently see "Bid-ask spread has reduced by x in the last two decades, therefore, HFT is good", but I've never seen a causal link explained, and a lot of other things have happened to the structure of the markets in that time.

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It's not exactly proven and would be very hard to prove. That said, nearly everyone thinks electronic markets are better than the alternative. Any electronic market will open itself to algorithmic trading, so what most people are actually worried about are unintended consequences of changes to the current system.

That and for most people who actually understand the mechanics what is going on, the current system seems way better for the vast majority of people than the one it replaced, so we have little reason to change.

I think it's pretty clear that HFT has a big role in this. If you look at the order books of US equities, most of the most aggressive orders on the book are ones that look like they come from high frequency strategies. Furthermore, if you believe that around 50% of trading volume is from HFT, and most of them are market making, then yeah, it lends one to conclude that the high frequency players are the ones who place more aggressive orders (which means, ones which reduce the spread) which gets them executions.

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