Sure you can harmonize other scales. How to name some of the chords may get tricky, but it's a legitamte way to develop chords. But I don't think of most basic chords as coming out of anytihing but the major scale. And there are two ways to look at them...
First, learn what intervals make up the basic chords. For example, a major chord is the 1, 3, and 5. A minor is 1, b3, and 5. Dominant 7 (mostly called a 7th) is 1, 3, 5, b7. Major 7 is 1, 3, 5. 7. Major 13 is 1, 3, 5, 7, 9 (2), 11 (4) and 13 (6) or all the notes in a major scale. Now you generally won't VOICE that chord using all seven notes, but that's what it is. Anyway, any good basic theory book will get you into this.
The other way to look at basic chords is to learn the harmonized major scale. Write out a C major scale. You'll get:
C D E F G A B C
Then stack the thirds on top, staying in the key of C. You'll get
E F G A B C D E
C D E F G A B C
Then repeat, staking the next thirds on top- you'll get:
G A B C D E F G
E F G A B C D E
C D E F G A B C
And one more time for:
B C D E F G A B
G A B C D E F G
E F G A B C D E
C D E F G A B C
Now if you analyze each of those stacks you'll find that the first chord (C E G B) is a major 7, as is the fourth (Fmaj7). The second, third, and sixth chords are minor 7 (Dmin7, Emin7, Amin7). The fifth chord (BTW, the fifth note is called the dominant note) is the only dominant 7 (1, 3, 5, b7). The seventh chord is weird- it's a minor 7 with a b5. Sometimes it's called a "half diminished" but I stick with the description that tells exactly what it is- "minor7 b5".
Now do this for yourself in all 12 keys. You'll find that there's this same progression of chords-
I and IV are major 7
ii, iii, and vi are minor 7
V is dominant 7
vii is minor 7 b5.
That's what people mean when they talk about a progression being "two five one"- In the key of C it's Dmin7 to G7 to Cmaj7. In the key of Eb it's Fmin7 to Bb7 to Ebmaj7.
Work this out for yourself on paper and in your head. THEN find those notes on the neck. This is the single biggest help for learning how chords and bass lines go together and how scale create chords.
jte