Why can't Windows XP handle newer SSL certificate versions?

OK, so looking at the example of mpql.net, we start with the SSL Labs analysis.

The problem appears to be that the server only supports elliptic curve cryptography (the various TLS_ECDHE_xxx suites) and, according to the MSDN articles Secure Sockets Layer Protocol and TLS Cipher Suites, Windows XP doesn't include any of the elliptic curve protocols. This is not related to the certificates per se, but to the way the web server is configured.

Firefox still works because it uses its own cryptographic library rather than using the SSL support built into Windows. Of course, if you were using a version of Firefox as old as Windows XP is, it probably wouldn't work either. :-)

certain websites cannot be accessed by Chrome and IE on WinXP due to incompatibility. However, Firefox apparently still does support Windows XP and can access those websites freely.

Chrome and IE use the CA store of windows while Firefox uses it's own independent store.

Why can't a simple community developer just create a "patch" for Chrome and IE if Firefox can support them?

It might be that some developer can do it, but why should somebody invest its time to support an OS which is dead and insecure just to support browsers which are no longer supported (Chrome will drop support end of 2015). If you feel this needs to be done anyway just do it. Otherwise, drop XP or at least use Firefox with it.

Solution for this problem in XP is installing KB3055973-v3 that adds support for TLS 128-bit & 256-bit Advanced Encryption Standard (AES) cipher suites. Google for this solution and beware it is natively for English OS version, otherwise you need to modify the installation files.