What is the relevance of price-time priority when each stock trades on many stock exchanges?

The majority of stock exchanges seem to arrange limit orders by price-time priority. This makes sense when each stock trades on only one stock exchange. If my limit order is the first in line on that stock exchange, I am really the first in line everywhere. This makes my first-in-line position meaningful.

However, if there are multiple stock exchanges trading the same stock, my position as the first in line on one stock exchange is less meaningful. Other market participants are able to jump the queue by placing their limit orders (at the same price as my order) on other stock exchanges, where they can be the first in line too. Given that there are so many stock exchanges (and hence many queues), the value of a limit order having price-time priority is greatly diminished.

What is the use of price-time priority in a market where each stock trades on many stock exchanges, considering that being the first in line on one stock exchange is almost meaningless?