I'm surprised to read this from a web designer.
You're right, it *is* the 21st century. Unlike in the last century, people aren't tied to using a Windows computer just to browse the internet anymore. Here in the 21st century, people browse the internet on all manner of devices - many of which are not supported by Adobe.
Can I install flash player on my iPod Touch? No. What about on my Android 2.1 phone? No. What about my friend's iPad? No. Or her iPhone? No. Can I install Flash player to the browser on my Kindle? No. What about all the Android 1.6 tablets that came out for Christmas? No. All of these things, and more, we browse the internet on - none of them can view Flash animations.
Using a proprietary plug-in is the thinking of the Dark Ages of the internet, it's certainly not "21st century". Apparently there were around 10 million iPads sold last year, so that's 10 million people who can't view your Flash websites, and no amount of saying "install the free Flash player" is going to change that.
I'm not saying that people shouldn't use Flash, I'm just highlighting the significance of how restrictive it is.