"No, I will not do IE6 support for you. You've trusted me to solve your problem in the best and most efficient way, and supporting IE6 would be a waste of your money (and my time)."
What your statement shows is not a understanding of how to solve the problem for such a client, but a rather a complete lack of understanding. Check the audience metrics, check the real problem being solved, and leave the grandstanding aside.
And for the record, I hate IE (6 or otherwise) and never want to support it. But I am not going let me personal dislike for a technology get in the way of helping a client solve a really important problem.
I guess we're fortunate in the sense that our software is important enough and expensive enough for top management to make IT allow a new browser just for us.
At the UK NHS, who exactly is the CIO, who exactly is mandating IE6 and only IE6? Can't that person be reasoned with? Overridden for the good of the NSH? Fired even? Just saying 'oh they're only allowed IE6' is giving up too early.
Many legacy web applications were built to work on IE6, and changing to a more modern browser can subtly, or completely in some cases, break these critical apps. It may be a pain for users when their favourite social media site looks rubbish or doesn't even run in IE6, but it's nothing compared to the finance department not being able to run their end of year reports because one screen in their GL system doesn't render properly in Firefox.
And it costs money firstly to test every single function of every single app that your business uses, and secondly to fix those places where it breaks (especially when it's an external vendor's application). That time and money can often be better spent doing one of the thousand other things that are making demands on the IT budget. Most companies are slowly making this transition, but unfortunately it's not just a case of saying "This browser it better/more secure than that one. Let's upgrade".
Governments, for all their posturing, love dealing with oligopolies, or even better, monopolies. They feel safer in making purchases from them. So around 2002, the UK government decided IT was so important, it was going to be done centrally for the whole NHS. They gave the job to Richard Granger, who then proceeded to divide England into five regions, each of which would have one vendor responsible for all software in the NHS for that region.
As part of that deal, they licensed MS Windows and Office. At that point, it became more expensive to use open source software than Microsoft's software. Then they standardised on IE6, and many vendors began using ActiveX.
When I ask departments to upgrade, they say they are going to do so, but first they need every single vendor to have signed off that their software works in the new browser. This will take a really long time.
Some will install a new browser for us, but this will actually result in a worse outcome for the clinicians because they have to use IE6 for most of their existing browsing, and then the other browser for us. They do not understand the point or the difference, and new browsers mean a different user interface, and they want to get on with treating patients rather than choosing between browsers.
The epilogue is that the UK government's national program wasted 12 billion pounds (about 20 billion US, ie the same amount the US government is also identically wasting to subsidise electronic health record purchases). And Richard Granger left the UK in disgrace.
He is now a consultant in... Australia :)
Instead you have to take the time to understand why they are requesting it and then communicate the tradeoffs to them at the appropriate technical level for their expertise.
In my experience, clients are usually pretty good at providing rationale for their requests, so my implied scenario is a bit of hyperbole. Often when the topic of legacy browsers comes up, it's because the client has been in the process of having a website made before, and is relieved to discover that the technology has moved forward since then.