  Strongly lensed explosive transients such as supernovae, gamma-ray bursts,
fast radio bursts, and gravitational waves are very promising tools to
determine the Hubble constant ($H_0$) in the near future in addition to
strongly lensed quasars. In this work, we show that the transient nature of the
point source provides an advantage over quasars: the lensed host galaxy can be
observed before or after the transient's appearance. Therefore, the lens model
can be derived from images free of contamination from bright point sources. We
quantify this advantage by comparing the precision of a lens model obtained
from the same lenses with and without point sources. Based on Hubble Space
Telescope (HST) Wide Field Camera 3 (WFC3) observations with the same sets of
lensing parameters, we simulate realistic mock datasets of 48 quasar lensing
systems (i.e., adding AGN in the galaxy center) and 48 galaxy-galaxy lensing
systems (assuming the transient source is not visible but the time delay and
image positions have been or will be measured). We then model the images and
compare the inferences of the lens model parameters and $H_0$. We find that the
precision of the lens models (in terms of the deflector mass slope) is better
by a factor of 4.1 for the sample without lensed point sources, resulting in an
increase of $H_0$ precision by a factor of 2.9. The opportunity to observe the
lens systems without the transient point sources provides an additional
advantage for time-delay cosmography over lensed quasars. It facilitates the
determination of higher signal-to-noise stellar kinematics of the main
deflector, and thus its mass density profile, which in turn plays a key role in
breaking the mass-sheet degeneracy and constraining $H_0$.
