Re: Certificate chain
"Thus if you run a web app, best to check it in all the major browsers.."
True but primarily for application issues.
For HTTPS configuration, running a test for that specifically (eg Qualys server SSL test) and actually understanding its results is best.
In both cases, if you follow standards, there is a good chance you won't have any problem.
Clearly, the DVLA (or subcontracted entity) didn't do this, which is a big fail.