  Probing the molecular gas reservoirs of z>~6 quasar (QSO) host galaxies is
fundamental to understanding the coevolution of star formation and black hole
growth in these extreme systems. Yet, there is still an inhomogeneous coverage
of molecular gas tracers. To measure the average excitation and mass of the
molecular gas reservoirs in the brightest z>6.5 QSO hosts, we combined new
observations of CO(2-1) emission with existing observations of CO(6-5),
CO(7-6), [C I], [C II], and dust-continuum emission. We reduced and analysed
the VLA observations of CO(2-1) in three z=6.5-6.9 QSO hosts -- the highest
redshift observations of CO(2-1) to date. By combining these with the nine
z=5.7-6.4 QSO hosts for which CO(2-1) has already been observed, we studied the
spread in molecular gas masses and CO excitation. Two of our three QSOs, were
undetected in CO(2-1), implying more highly excited CO than in the z=6.4 QSO
J1148+5251. We detected CO(2-1) at $5.1\sigma$ for our highest-redshift target,
J2348-3054, yielding a molecular gas mass of $(1.2\pm0.2)\times 10^{10}\,
\mathrm{M}_\odot$. This molecular gas mass is equivalent to the lower limit on
the dynamical mass measured from resolved [C II] observations, implying little
mass in stars or neutral gas within the [C II]-emitting region. On average,
these QSO hosts have far higher CO(6-5)-, CO(7-6)-, and [C II] vs CO(2-1) line
ratios than local AGN hosts; with a mean CO(6-5)-to-CO(1-0) line luminosity
ratio of $r_{6,1}=0.9\pm0.2$. Our new CO(2-1) observations show that even at
780 Myr after the Big Bang, QSO host galaxies can already have molecular gas
masses of $10^{10}$ M$_\odot$, consistent with a picture in which these z>6
QSOs reside in massive starbursts that are coevolving with the supermassive
black holes. Our results imply the presence of extremely dense and warm
molecular gas reservoirs illuminated by strong interstellar radiation fields.
