No, I don't agree. Context is important.
I'm sure I've felt the emotion engendered by the Last Post at various Anzac days, funerals and memorials as much as anyone, and it's a powerful part of all that.
Tim Vine recalling funny occurrences from his school days doesn't detract from it, in my view, and I'm sure we've all had times where we found something funny at an inappropriate moment and had to choke back the sniggers. Well, I know I have, anyway.
Comedians often pull the piss out of sacred cows, and I'm glad we live in a society where you can do so without being targeted (literally) for saying the wrong thing about some religious figure or cultural icon.