Is it possible to make decimal computer using binary technology?
Actually the ENIAC, considered to be the first large scale general purpose programmable electronic computer, used a scheme almost identical to what you propose.

It used ten-position ring counters to store digits in ten's complement representation.  Each digit used 36 vacuum tubes, 10 of which were the dual triodes making up the flip-flops of the ring counter (corresponding to the ten transistors in your proposal -- actually you would need a minimum of twenty transistors so you could have ten flip-flops to store the state, corresponding to the ENIAC's dual triodes).

Arithmetic was performed by counting pulses with the ring counters and generating a carry if the counter wrapped around.  ENIAC had twenty ten-digit signed accumulators.

So your idea goes back almost 70 years.

Other notable decimal computers were the Harvard Mark 1, the IBM 650 and successors 7070 series, and the UNIVAC Solid State (competitor to the 650).