  We present a first complete implementation of an effective-one-body (EOB)
model for extreme-mass-ratio inspirals (EMRIs) that incorporates aligned spins
(on both the primary and the secondary) as well as orbital eccentricity. The
model extends TEOBResumS-Dal\'i for these binaries by (i) recasting
conservative first-order gravitational self-force (1GSF) information in the
resummed EOB potentials; (ii) employing a post-Newtonian (PN)
$3^{+19}$PN-accurate (3PN comparable-mass terms hybridized with test-particle
terms up to 22PN relative order) expression for the gravitational-wave flux at
infinity; (iii) using an improved implementation of the horizon flux that
better approximates its test-mass representation. With respect to our previous
work [Phys. Rev. D 106 (2022) 8, 084062], we demonstrate that the inclusion of
the $3^{+19}$PN-accurate $\ell=9$ and $\ell=10$ modes in the flux at infinity
significantly improves the model's agreement with second-order accurate GSF
(2GSF) circular waveforms. For a standard EMRI with mass ratio $q \equiv
m_1/m_2 = 5 \times 10^4$ and $m_2 = 10 M_\odot$, the accumulated EOB/2GSF
dephasing is $\lesssim $ rad for $\sim 1$ yr of evolution, which is consistent
with the standard accuracy requirements for EMRIs. We also showcase the
generation of eccentric and spinning waveforms and discuss future extensions of
our EOB towards a physically complete model for EMRIs.
