I think you're right that most folks were likely ignorant as to the difference.
OTOH, the "romantic notion of olden times" really didn't work for Takeda. He was kicking around teaching the sword, but making very little money - nobody was interested in "olden times" martial arts.
Having a loan on his house, he needed to make some money - so he started teaching jujutsu. Mostly, he taught to police or military in the towns he passed through, since they made up the bulk of the people who could actually afford his very substantial fees. Those types weren't very romantic - they were more practically oriented.
Right - and students came to him because he offered the skills and power of the bushi of olden times. There were certainly other types of training, more modern and more practical, that they could have sought out.