Why are analytic signals so important in Time-frequency analysis?
Assuming time-frequency aims a providing a separation (at least visual) between signal components, the main reasons could be:


for quadratic distributions, which tend to yield interference  between components, "cancelling" at negative frequencies reduce the quantity of components that can interfere.
for linear distributions, the filter bank formalism, and especially the down-sampling operators, is simplified, reducing the impact of aliasing errors.


For real signals, the Hermitian symmetry yield that "no information" is lost in the analytic form, a complex-valued function that has no negative frequency components, and it is easy to go back to real.

However, this is not so simple in practice. 


If the analytic signal constructed from a wide-sense stationary (WSS) real signal is always proper, this may not be the case for non-stationary signals, as underlined in Stochastic time-frequency analysis using the analytic signal: why the complementary distribution matters. 
Moreover, computing the analytic signal on discrete, finite-length data is often only approximate, especially for real-time applications. Hence the energy in the negative frequencies is rarely zero.
Extending the concept of analyticity beyond 1D is not evident, and several design still exist.