We could be wildly speculative.
But, consider that among all of the things we know about Takeda, nowhere do we hear of Takeda being a liar.
Not a liar, of course, but maybe he wasn't telling all the truth or all the relevant data.
If we read Tokimune's bio of his father, we find Tanomo not as teacher of martial skill but
Sokaku studied under Chikanori Hoshina (aka Tanomo Saigo) who was an assistant priest of the Nikko Toshogu Shrine in Mt. Futara. He studied the secret mind reading technique of aiki and acquired various super-human powers such as an unyielding spirit, clairvoyant power, and prescience.
his musa shugyo years around Japan.
Sokaku was sent by his father to Tanomo, for being tamed and for making him a priest, but after some time Sokaku (a teenager) escaped and tried to join Takamori Saigo troops. Musa shugyo stars.
Perhaps, the entirety of internal skills were not from Tanomo, but there are indications that something was taught. Something which Takeda defined as his "aiki" later on.
I think about Tanomo as the cherry on the cake, not as what the cake is made of.