Here's how I would think of the tune. Might not be the same as the rest of y'all but for my two cents...
The chord changes are A D E, which indeed looks like a I-IV-V in the key of A. In this case though, that isn't correct. It's actually a I-VII-IV in the key of E, which is also known as e mixolydian. We arrive at e mixo by taking the 7th chord tone of the major scale (in this case D#) and dropping it 1/2 step. (If the song was in e major the progression would go E - D#dim - A.)
Cork to answer your scale selection question, you can play A major over the changes but it'll sound square and very much like you aren't following the song. The notes will all be correct but you'll be playing them at the wrong time. (More on that below.)
A minor pentatonic won't work for the reasons below. (Note that these are basic reasons and if you're a good enough improviser you can get away with anything.) (*disclaimer)
- Over the E major chord (E G# B) you will be playing A C D E G. The A and G in your scale will clash with the 3rd of the E major. (G as a b3 of the chord can work, but it works much better in a different context that I discuss below.)
- Over the D major chord (D F# A) you will be playing A C D E G. The G in your scale will clash with the 3rd of the D major.
- The A major chord is least problematic, but the half step between the C in the scale and the C# in the chord is still an issue. In this case though it's a blue note, which you can get away with if you understand resolution. (Read that disclaimer again )
Clashing isn’t always a bad thing. Dissonance is important in music. But the dissonance has to properly resolve or it just sounds wonky and out of key. Suffice it to say that the a minor pentatonic is not a good "one size fits all" scale for this tune.
So you would instead play E mixolydian, which is enharmonic with A ionian. (Hence my "It's more about how the notes are used" comment.)
If you play this in A Ionian you will tend to resolve to A. However since this song is in E you want to resolve to your I chord...so even though the note selection is identical, "thinking" in e will (hopefully) result in resolving to the correct note(s) at the correct time, creating happy noises.
E mixo: E F# G# A B C# D
Note that the chord tones for all of the song's chords are in this scale (E G# B / D F# A / A C# E)
In my personal opinion that tends to sound a little square, so I play e major pentatonic and avoid that pesky 7th scale tone entirely...which to Russell's suggestion is enharmonic with C# minor pentatonic:
e f# G# B C#
For fun I'll throw in some e minor pentatonic just to create a bluesier feeling.
E G A B D
*Disclaimer: Strictly speaking a lot of these examples above involve playing b3rds and 4ths against major thirds, which is super common in blues and rock. The problem isn't really note selection but rather how the dissonant notes resolve to chord tones, and in the case of playing A minor pentatonic unless you're thinking in e, you won't resolve appropriately. Thus it will sound lost, dissonant, and likely that you are playing in the wrong key.