  Although evidence of inter-valley attraction-mediated by phonon or
topological fluctuations is accumulating, the origin of superconductivity in
the flat-band quantum moir\'e materials remains an open question. Here, instead
of attempting to pinpoint the origin of the superconductivity, we aim at
identifying universal properties of moir\'e flat bands that shall emerge in the
presence of inter-valley attractions. We show that by matching the interaction
strength of inter-valley attraction with intra-valley repulsion, the flat-band
limit becomes exactly solvable. Away from the flat-band limit, the system can
be simulated via quantum Monte Carlo (QMC) methods without sign problem for any
fillings. Combining analytic solutions with large-scale numerical simulations,
we show that upon increasing temperature, the superconducting phase melts into
a bosonic fluid of Cooper pairs with large/diverging compressibility. In
contrast to flat-band attractive Hubbard models, where similar effects arise
only for on-site interactions, our study indicates this physics is a universal
property of moir\'e flat bands, regardless of microscopic details such as the
range of interactions and/or spin-oribt couplings. At higher temperature, the
boson fluid phase gives its way to a pseudo gap phase, where some Cooper pairs
are torn apart by thermal fluctuations, resulting in fermion density of states
inside the gap. Unlike the superconducting transition temperature, which is
very sensitive to doping and twisting angles, the gap and the temperature scale
of the boson fluid phase and the pseudo gap phase are found to be nearly
independent of doping level and/or flat-band bandwidth. The relevance of these
phases with experimental discoveries in the flat band quantum moir\'e materials
is discussed.
