According to the current paradigm of circumstellar disk evolution, gas-rich primordial disks evolve into gas-poor debris disks compose of second-generation dust. To explore the transition between these phases, we searched for $\mathrm{<mrow\; data-mjx-texclass="ORD">12</mrow>}$CO, $\mathrm{<mrow\; data-mjx-texclass="ORD">13</mrow>}$CO, and C$\mathrm{<mrow\; data-mjx-texclass="ORD">18</mrow>}$O emission in seven dust-rich debris disks around young A-type stars, using ALMA in Band 6. We discovered molecular gas in three debris disks. In all these disks, the $\mathrm{<mrow\; data-mjx-texclass="ORD">12</mrow>}$CO line was optically thick, highlighting the importance of less abundant molecules in reliable mass estimates. Supplementing our target list by literature data, we compiled a volume-limited sample of dust-rich debris disks around young A-type stars within 150 pc. We obtained a CO detection rate of 11/16 above a $\mathrm{<mrow\; data-mjx-texclass="ORD">12</mrow>}$CO J=2$-$1 line luminosity threshold of $\sim 1.4\times 104$ Jykms$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$pc$2$ in the sample. This high incidence implies that the presence of CO gas in bright debris disks around young A-type stars is likely more the rule than the exception. Interestingly, dust-rich debris disks around young FG-type stars exhibit, with the same detectability threshold as for A-type stars, significantly lower gas incidence. While the transition from protoplanetary to debris phase is associated with a drop of dust content, our results exhibit a large spread in the CO mass in our debris sample, with peak values comparable to those in protoplanetary Herbig Ae disks. In the particularly CO-rich debris systems the gas may have primordial origin, characteristic of a hybrid disk.