This is for pedagogical reasons. In general, root-position chords are the most common, and even when inversions are used, it's usually fine to substitute with a root-position voicing, whereas replacing a root-position chord with an inversion can create an unstable or even unpleasant sound if you don't give due consideration to the harmonic context.

If there is an inversion that you are trying to play and can't find on a guitar site, it's probably because there isn't a good open voicing for it. Even fairly basic guitar chord charts usually include the open inversion chords D/F♯ and C/E, for example, because these are relatively easy to play and are common in guitar music. For a chord like E♭/G, there isn't really a good open voicing; instead you will play a generic shape for "major triad in first inversion," and figuring that out requires some theory knowledge that is readily available elsewhere on this site.

Especially when those same sites show 'scales' which include all available notes in a certain position - those below and above the root - which actually aren't part of a 'proper' scale (starting/finishing on root).

Seems like two differing concepts.

They are different concepts. A scale doesn't have a "bass note" in the way that a chord does. It has a root note, but this is a theoretical abstraction that allows us to give specific names to the different modes (E lydian, D locrian, etc.). There is no requirement that every instance of a scale contain the root note as the lowest note in the melody—otherwise music would sound strange indeed!

On the other hand, a D/F♯ chord isn't a D/F♯ chord unless the lowest note being played (by whomever is playing it; it could be a bassist or pianist instead of you) is an F♯.

This is for pedagogical reasons. In general, root-position chords are the most common, and when inversions are used, it usually sounds just fine to substitute with a root-position voicing (especially if you have a bassist or other instrument covering the bass notes). On the other hand, replacing a root-position chord with an inversion can create an unstable or even unpleasant sound if you don't give due consideration to the harmonic context. This is especially true of second-inversion chords, such as your example of an A chord with the low E string ringing out—namely, A/E. In an A major context, this chord doesn't sound "final." It sounds like it should go somewhere, as in the following progressions:

• The so-called grand cadence: A | D | A/E | E | A
• Something a little more colorful: A | A7/C♯ | D | B7/D♯ | A/E | Fdim7 | F♯m7 | B7 | D | E7 | A

If there is an inversion that you are trying to play and can't find on a guitar chord chart, it's probably because there isn't a good open voicing for it. Even fairly basic guitar chord charts usually include the open inversion chords D/F♯ and C/E, for example, because these are relatively easy to play and are common in guitar music. For a chord like E♭/G, there isn't really a good open voicing; instead you will play a generic shape for "major triad in first inversion," and figuring that out requires some theory knowledge that is readily available elsewhere on this site.

Especially when those same sites show 'scales' which include all available notes in a certain position - those below and above the root - which actually aren't part of a 'proper' scale (starting/finishing on root).

Seems like two differing concepts.

They are different concepts. A scale doesn't have a "bass note" in the way that a chord does. It has a root note, but this is a theoretical abstraction that allows us to give specific names to the different modes (E lydian, D locrian, etc.). There is no requirement that every instance of a scale contain the root note as the lowest note in the melody—otherwise music would sound strange indeed!

On the other hand, a D/F♯ chord isn't a D/F♯ chord unless the lowest note being played (by whomever is playing it; it could be a bassist or pianist instead of you) is an F♯.