This thing that Chris Li quoted captures another important part of it:
Chief among everything he had to offer, Takeda was selling a romantic notion of olden times to late-Meiji / Taisho / early Showa students. Because people wanted to be in touch with that and they also weren't widely knowldgeable in what it actually was.
I am betting that if you interviewed typical budoka in Japan from those periods, you would not find many of them (who were not actually involved with one of the old systems at the time) had any idea what the traditional training structure of a classical jujutsu dojo was. The idea that the teacher threw the students around most of the time would probably seem likely to them.
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.