What is the difference between a collection of Turing Machines and a family of Circuits?
What you describe is essentially Turing machines with advice, the advice for length $i$ being simply the description of $T_i$. It is a classic result that the two models are equivalent in the case of poly-time TMs and poly-sized circuits, that is, both produce the same class $\mathsf{P}/\mathrm{poly}$. If the description length of $T_i$ is allowed to be arbitrary, there is also no computational power to be gained since you can then express all Boolean functions for each length (just as with circuits).