  We present multichannel quantum defect (MQDT) models for highly excited
$^{174}$Yb and $^{171}$Yb Rydberg states with $L \leq 2$. The models are
developed using a combination of existing literature data and new,
high-precision laser and microwave spectroscopy in an atomic beam, and
validated by detailed comparison with experimentally measured Stark shifts and
magnetic moments. We then use these models to compute interaction potentials
between two Yb atoms, and find excellent agreement with direct measurements in
an optical tweezer array. From the computed interaction potential, we identify
an anomalous F\"orster resonance that likely degraded the fidelity of previous
entangling gates in $^{171}$Yb using $F=3/2$ Rydberg states. We then identify a
more suitable $F=1/2$ state, and achieve a state-of-the-art controlled-Z gate
fidelity of $\mathcal{F}=0.994(1)$, with the remaining error fully explained by
known sources. This work establishes a solid foundation for the continued
development quantum computing, simulation and entanglement-enhanced metrology
with Yb neutral atom arrays.
