So, is the only way for us to get the Game Packs to download them from the EA/Origins site?
Unfortunately, until something is officially said, it looks like you will have to get the Game Packs from EA/Origin. Note that some places like Amazon do allow you to buy the Game Pack, but in that instance it gives you a code to redeem on Origin.
Are they likely to be changing that in time, and make them available through stockists as well, or will downloading them online be the only way of getting the content?
Most likely, unless EA changes their mind, it will be download only. I wouldn't be surprised if they didn't 'sell them' in a store, but it would be like the Sims 3 worlds, then, and it would be an empty box with a download code.
Are all the Stuff Packs and Expansion Packs likely to be only available in this way as well?
Expansion Packs will probably not change; they will be available both in store or online, though the online version may offer a few minor bonus items.
To be honest, I am a little uncertain as to whether Stuff Packs will even remain around, especially if the Game Packs take off.