> Do exchanges give us insight into what their order books actually look like, instead of just quoting the last price where a buyer and seller matched?

They do. I haven't seen depth info past the best bid/offer (maybe exists? I'm not an expert), but I see the bid and ask prices with sizes in my broker's app in realtime. "Last price" is just one useful bit of information.

The theory behind showing the last price is that an actual matched transaction feels less likely to be "wrong" (over short periods) than bid/ask in illiquid cases. Sometimes there really just aren't people on one side ready to buy/sell, and the ”$1 $1 $1” would say something less realistic, not more.

Good question though, dunno why anyone would downvote.