would universal use of arbitrage bot stabilize the market? 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.