Hi David -
Well your 'coder' is right... AS3 is not going away anytime soon - but sites/browsers/devices which support Flash are quickly sliding off into the ether. Also - please note I am no expert on ANYTHING - but, like everyone, have both opinions and a belly-button.
It's not so much that Flash is 'seamless' - it's a wall-garden of execution that has been ported to run on several platforms. The browser just hands off whatever is your request and the flash-execution-code for that environment runs it. HTML5 is supposed to replace Flash to the extent that platforms and devices which are "HTML5 compliant" should be able to take the same input and prodice the same results. However, as best I can tell, it is not a perfect world and it's still relatively early in the HTML5 world.
At one point all you had to worry about was Internet Explorer, FireFox, Opera, Safari and other desktop browsers. Now there're mobile devices in the Android and iOS world which have incredible screen-size limitations, limited processing power, limited memory resources, little or no opportunity for 3rd-party plugins, and so on.
From my perspective, it makes more sense to go with HTML5 because, Flash is, as they say, "So 20th century".
Depending on the nature of your website [are your pages all animation and graphics - or spreadsheets - or textual with images, one other possible solution you might want to look into is an "Application Server" - like JBoss - where you can write all your 'business logic' in Java (and other languages like JavaScript, PHP, etc).
Yet another approach might be to go with something like Drupal or Wordpress or Joomla! - or any number of such environments. They can do some incredible stuff and it's completely customizable.
Well - that about drains my brain-jiuce ono the subject. GOOD LUCK!