  This paper considers the discretization of the Stokes equations with
Scott--Vogelius pairs of finite element spaces on arbitrary shape-regular
simplicial grids. A novel way of stabilizing these pairs with respect to the
discrete inf-sup condition is proposed and analyzed. The key idea consists in
enriching the continuous polynomials of order $k$ of the Scott--Vogelius
velocity space with appropriately chosen and explicitly given Raviart--Thomas
bubbles. This approach is inspired by [Li/Rui, IMA J. Numer. Anal, 2021], where
the case $k=1$ was studied. The proposed method is pressure-robust, with
optimally converging $\boldsymbol{H}^1$-conforming velocity and a small
$\boldsymbol{H}(\mathrm{div})$-conforming correction rendering the full
velocity divergence-free. For $k\ge d$, with $d$ being the dimension, the
method is parameter-free. Furthermore, it is shown that the additional degrees
of freedom for the Raviart--Thomas enrichment and also all non-constant
pressure degrees of freedom can be condensated, effectively leading to a
pressure-robust, inf-sup stable, optimally convergent $\boldsymbol{P}_k \times
P_0$ scheme. Aspects of the implementation are discussed and numerical studies
confirm the analytic results.
