Ask HN: Why does Apple make my enter my password when downloading free apps?

Once you've downloaded an app under an Apple ID, that app is marked as downloadable under other devices signed into that ID. Whatever cloud data is associated with that app is sometimes also downloaded with that app on other IOS devices with the same account. In general, this is a convenience win.

So the password entry in this case is not so much an authorization to put the app on the device, but more permission to associate the app and possible data sync with the ID.

I know this because as a parent managing multiple kids IOS devices, I elected to manage family IOS devices all under one ID for simplicity. With the recent IOS 7 update and increased automated cloud syncing with various apps (including the default Apple apps), this has become a more difficult configuration to manage. There are more options to have to check, and I'm still not sure there are options in enough places to manage the kids devices as I did pre-IOS 7. It ends up be a subtle push to setup separate IDs for each individual.

Consumer conditioning, it eliminates a psychological barrier to getting paid apps. If normally you download free apps and then one time you get a paid app and you have to enter a password then it makes you think twice and increases the risk you'll reassess and stop the sale. The way it is now the habits involved in getting paid or free apps are the same. It's apple simplifying things for you mentally even though it takes more of your time.

It's at least partially because apple doesn't prompt for a password on in app purchases for 20(ish) minutes

Speculating here, but probably data collection, usage tracking. App makers probably want to send you updates/emails on why you should buy the pro version of the app etc, but I think your question might have be rhetorical.