I don't see "twee" as implying particularly old-fashioned style. More a matter of being sickly sweet, and that can come in an ultra-modern context just as often.

"Old timey" seems to me to be analogous with the similar folksiness of "Grand Ole Opry". Maybe a phoney folksiness, but not necessarinily.

I wouldn't call those birchbark canoes "twee". Phoney, fake, ersatz, but not twee. But I'd not mind having one!