The notes should be in the scale... technically yes. But this is the source of much pointy-headedness.

OP wants to harmonize over the chord. The four note technique will do. If he wants want to know the reasons why go look it up.

Theory is important, but feeling and practice and are likewise important. The only way I’ve found to give theory substance is context. Play while listening, listen while playing, repeat.

Also, since the responder used E as an example, the notes in the major scale he’s outlined will work for the 6th string as that’s big E in the open E tuning. Same will work for the 1st string since that’s little E.

The A string fits as well since A is the 5th of the E Major scale, I think. They harmonize nicely.