I think it could be done on the Amiga, but you'd probably need quite a bit of ram to do it.
Following Archer Macleans example for Jimmy Whites Whirlwind Snooker, when you go to take a shot in that game, the whole time the snooker cue is moving after you let it go, is when its doing all its calculations.
Depending on just how big the screen is, and how many bitplanes you're aiming for (the objects in Angry Birds aren't that colourful), you could opt to do a similar system, where after you release your bird, theres a pause as it precalculates everything, and draws all the relevant screens to maintain a decent speed and no ingame slowdown.
Obviously when the Bird leaves the catapult is simple blitting until it interacts with something, and then it precalculates everything.
The only time this wouldn't work is if you use an exploding Bird, because obviously the precalc routines would have no clue when you'd activate it.
Or, simply pause whilst it does its calculations.
Lets not be coy here, whilst Angry Birds is a simplistic concept, its running on smart phones that are ten times more powerful than an Amiga A500, i've seen Angry Birds drop frames on lesser smart phones.
With some programming ingenuity, Angry Birds is definately possible, and quite a close conversion would be possible, but, its personally not the game i'd aim for, because from a novice gamers perspective, they think Angry Birds is simple, they would have no clue in the amount of work needed to actually realise something like this on the Amiga.