If each bitcoin exchange ran a bitcoin-arbitrage bot (let's say the same one)themselves, would that quickly eliminate the spread, and therefore the opportunity for arbitrage?
Yes, and I recommend researching market making algorithms.
The problem is that verification and confirmation times are high, so the spreads or at least trade fees will remain wide until that problem is fixed.
Yes, it's by design, but it's an economic problem, and wide spreads are the economic result.
No it would not. Due to fees involved with transferring assets between exchanges, and exchange commissions, inter-exchange spreads would still exist. Everyone running bots would at best lower spreads, not eliminate them.
There are already bots running on every market you can imagine. Because exchange operators are highly technical, most exchanges offer API's from the beginning.