  SrTiO$_3$-based conducting interfaces, which exhibit coexistence of
gate-tunable 2D superconductivity and strong Rashba spin-orbit coupling (RSOC),
are candidates to host topological superconductive phases. Yet,
superconductivity is usually in the dirty limit, which tends to suppress
nonconventional pairing and therefore challenges these expectations. Here we
report on LaAlO$_3$/SrTiO$_3$ (LAO/STO) interfaces with remarkably large
mobility and mean free paths comparable to the superconducting coherence
length, approaching the clean limit for superconductivity. We further show that
the carrier density, mobility, and formation of the superconducting condensate
are controlled by the fine-tuning of La/Al chemical ratio in the LAO film.
Interestingly, we find a region in the superconducting phase diagram where the
critical temperature is not suppressed below the Lifshitz transition, at odds
with previous experimental investigations. These findings point out the
relevance of achieving a clean-limit regime to enhance the observation of
unconventional pairing mechanisms in these systems
