Im on the 737-800. The EFIS system on that plane is certified for Approach Navigation. Basically, the Map Display is as good/better than any conventional navigation display. If you in a 10 mile scale or less, and the airplane touches the magenta line, you're good.
For takeoff, we check airplane symbol touches the runway and headings are aligned with the takeoff runway, you're good to go.
During preflight, we do check other traditional stuff. but, those checks strike me as "Ancestor Worship".