What would the goodness of fit statistics look like for a scenario where the E model was preferable to the ADE model?
If the phenotype has absolutely no genetic influences, the E (or CE) model will fit best on a LRT, and likely all other model fit statistics.
What does “fitting best” entail- lowest AIC, and…?
Lowest AIC, lowest BIC. If you want to consider RMSEA, CFI, and TLI, those are fine too. In general, nested model comparisons is the way to determine best fitting.