Multi-watt continuous-wave (CW) operation of tetragonal rare-earth calcium aluminate Yb:CaLnAlO4(Ln=Gd,Y)) crystals in plano-plano microchip lasers was demonstrated with an almost quantum-defect-limited slope efficiency. Pumped at 978 nm by an InGaAs laser diode, a 3.4 mm long 8 at. % Yb:CaGdAlO4 laser generated 7.79 W at 1057-1065 nm with a slope efficiency of η=84% (with respect to the absorbed pump power). An even higher η=91% was achieved with a 2.5 mm long 3 at. % Yb:CaYAlO4 laser, from which 5.06 W were extracted at 1048-1056 nm. Both lasers produced linearly polarized output (σ-polarization) with an almost circular diffraction-limited beam (Mx,y2<1.1). The output performance of the developed lasers was modeled, yielding an internal loss coefficient as low as 0.004-0.007 cm-1. In addition, their spectroscopic properties were revisited.