I live in Roppongi. For me, there are 3 wards to Tokyo. Shibuya and Shinjuku still have applications pending. :-) The rest are all inaka for intents and purposes.

To the OP: A Japanese family will offer courtesy to those they wish to show respect and friendship. Not staying with them is by no means an affront, but good form in Japan dictates a small white lie such as hotel is prepaid or similar so as to make everyone feel the extended offer was amazing but quite impossible to accept.

If you do not or cannot stay with them for a day or two (which may be an amazing cultural experience by the way), invite them to meet you for a meal and when they insist on paying make a big fuss abut sharing the bill but concede at the end and they will feel they have done their bit.