Web Developers have learnt from the past, there will never be a supported code that will be dependant on a specific version again.
Cross-compatibility and Browser Independence is a main focus that hasn't been in the past. Most websites are not locked into a particular browser, so there are more options if things go pear-shaped in a particular browser. If for example Firefox drops XP support and there is a bug with the old version, the customer can change to Chrome until another solution is put in place.
IE6 was the exception, because it was too difficult in many codebases to update it for compatibility beyond IE6 in the short term, for time(=money) reasons. As soon as the codebases were updated (or the solution replaced) to work beyond IE6, IE6 was kicked right out the door. IE6 didn't stay king because so many people loved that browser so much that they didn't want to change, it was because they HAD to keep using it for some reason. It is not uncommon for companies still relying on IE6 to have Firefox installed for general web browsing and IE6 only for the specific app they need. You can bet your ass they have retirement plans on how to eventually get off IE6 (& now also XP) altogether.
Unsupported code (eg: unmaintained websites) that won't work with new versions - Yes that is inevitable.
Supported code - No.
If it is a supported codebase - The web developer's solution would be to update it to work with the new version, not make it work with the old. If that means that it will break compatibility with the old version, then so be it, it is industry practice not to support unsupported software.
It's worth pointing out that Mozilla & Google are not supporting XP - They are supporting their browsers. If there is a problem in XP, they are not going to help you with it.