Transmission of sampled signal
You are right. If you really want to send the sampled signal $$x_s(t) = {\delta}_{T_s}(t) x_c(t) = ( \sum_k \delta (t- k T_s) ) x_c(t) = \sum_k x_c(k T_s) \delta (t-k T_s)$$ then this weighted impulse train constitudes an infinite amount of (shifted replicas of base period) bandwidth in frequency domain.

Therefore this signal is not transmitted anywhere... Indeed the impulse sampling cannot be realized in practice either. It's just a mathematical model which provides a quantitative description of the conversion process from a continuous time-signal into a discrete-time sequence whose samples have values equal to the continuous-time signal at those sampling impulse instants.

The sampling process is used as the initial stage of conversion from continuous to discrete time domains and it's the sequence of those samples which are processed later on for whatever purpose you have.