Why foreigners are forced into some hotels isn't as clear as you suggest. They may be able to rent varieties of short or long term accomodation in many places; however, they are restricted in some areas and hotels. For example, foreign nationals aren't allowed to live in some parts of China and/or some parts such as villages where farmers live, and they aren't permitted to check into some cheap hotels that may accomodate local budget travelers. This may indicate the fear of outside influences that aren't desired for the sake of the local harmony.

Your South Africa comment sounds like from a doctor whom I have known before. He kept on doubting that forums could achieve a lot more than what he assumed they would. The SA "regime change" came with the help of man who wouldn't have been able to get a hotel room next to the white man. In my case, I just want to get a reasonably priced room with my colorful family which you don't seem to be able to either understand or address appropriately.

Foreigners with visas (I'm not sure what the situation is for someone with an actual 'green card'; they are few and far between in China) must stay at hotels that have the capacity to register the foreigner with the local PSB. Routinely, that is now an on-line transaction but it requires that the hotel have the equipment AND the permission to book foreigners into their hotel. All 4 and 5 star hotels have that capability and many, if not most, 3 star hotels.
The Chinese government would say that this is to protect their 'foreign guests', but of course it is to keep track of them.

If South Africa had required that blacks could only reside in 4 & 5 star hotels, I doubt that Mandela would have achieved much traction in his 'regime change'.