The job of a market maker is to supply liquidity at a price/risk tradeoff that is reasonable to them, subject to the information available to them. If there are multiple exchanges, and someone trades with them on one exchange, then the set of information available to them has changed (specifically, their knowledge of the supply/demand balance for a particular stock has changed). It's only natural that they will want to change their prices in response.
Now, we could change legislation to either (a) go back to having a single exchange or (b) restricting the ability of market makers to move their quotes on one exchange if they trade on another. But that won't necessarily result in a better deal for non-market makers, because instead of quoting 20,000 shares split across 4 exchanges, the market maker now quotes 5,000 shares on 1 exchange.
The benefit is that all market participants have a more accurate idea of the true liquidity available in the market. The disadvantage is that you have removed the element of competition between exchanges, so the exchange is no longer incentivized to offer low fees and keep improving its service.