So first off, the case $T(x)=x$ on the whole line is nonphysical, so you should first forget about that. A temperature distribution on an unbounded domain will go to some constant at infinity in a reasonable physical situation. Intuitively speaking, you can't exchange heat with infinity.
On a bounded domain, you have boundary conditions. For example, if there's no bulk heat input but you have boundary conditions $T(0)=0,T(1)=1$ then indeed $T(x)=x$ is a steady state. But the point is that there actually is heat flow here: heat is flowing into the rod on the right at a certain rate and leaving the rod on the left at exactly the same rate. The temperature profile isn't changing despite this flow.
On the other hand, if you require that no heat be conducted on the edges either, then the suitable boundary condition is $T_x=0$ on the boundary. A nonconstant linear profile does not satisfy this boundary condition. If you try to prepare a linear profile in this situation, intuitively your profile will really be something which looks linear and then rapidly flattens out near the boundaries. The little regions where the transitions from $T'=1$ to $T'=0$ occur will have nonzero $T''$, so heat will start to flow and you'll get a flat temperature profile in the long run.