Tl;DR: I think with the right tools and practises you could create a situation where automaticaly testing cross-browser is not really needed.
I think today most cross-browser issues can be found by Application Monitoring Tools like New Relic. Observing the behavior, worksflows and errors of the end-users for each browser. This is sometimes called "Testing in production, the safe way".
There are differences in rendering between the major browsers, for example in the CSS defaults. That is why the developers probably should use a CSS Reset. Just assuming everything will look and feel the same in all browsers is a myth, even if the DOM is probably the same it might look different or even cut off parts.
Personally I have been making business software for a couple of years, most companies standardize on a single browser. My teams have been testing versus a single browser for years, even if some users might use a different browser. Can't remember it breaking, but users have a fallback use our main supported browser and report a bug.
I would check what browsers my userbase is using. Then check if there are known issues with those browsers currently.
Asking the developers to use a webbased UI and CSS framework (one that is tested cross-browser) should lower the risk even more. Try to make the cross-browsers testing someone elses problem.