Personally, I've always thought the goal was to provide a good UX for every user, regardless of their browser's features and modernity. This doesn't always equate to identical page renderings and functionality, though.
At work I'm being asked to make sure pages render identically in every browser, which means either dumbing down design to not use the newest features, or loading a ton of polyfills in legacy browsers to achieve "browser transparency," which as far as I see, hurts UX because they have longer page loads for styles and features that don't affect the page's usability.
Taken to its extreme, this would result mean design should cater to the lowest common denominator with no JS (some people don't use JS) and that any features beyond this should be excluded in the name of uniformity.
Am I wrong in thinking that identical UX should not be a goal, but rather good, consistent UX to the best capabilities of the browser?
When developing for multiple browsers, you cannot provide an identical experience for every browser. After all, someone could try to access your site via Lynx, and you're clearly not going to design your UX to ensure that someone who is using Lynx will have an identical experience as someone who is using the latest version of Chrome.
The first question to be answered is to determine which browsers your users are using. If you have a current version of the application, you can answer this question via log data. If you don't have a current version of the application, you can try to approximate an answer to this question by using general web usage statistics or by surveying representative target users for your application.
With this information, you can then make an informed decision about how to target your efforts. As you have already identified, there is a trade-off between identical support for all supported browsers and having a great user experience for most of your users. With the data about what browsers are actually being used by each percentage of your users, you can then identify exactly the trade-offs that you would make by having an identical experience across all browsers or by having a great UX that targets most of your users and a degraded experience for a subset of your users.