Why does the formula for an augmented 7 chord contain a flat7 note, not a major 7th note?
This is how the notation is defined. A "7" by itself means a ♭7; a major 7 must be notated explicitly.
A = A major
A7 = A major, plus a ♭7
A△7 = A major, plus a major 7
A- = A minor
A-7 = A minor, plus a ♭7
A-△7 = A minor, plus a major 7
A+ = A augmented
A+7 = A augmented, plus a ♭7
A+△7 = A augmented, plus a major 7 (also A△7[♯5])
Diminished chords are an exception.
Adim = A diminished
Adim7 = A fully diminished seventh (also Ao7)
A-7(♭5) = A half-diminished seventh (also Aø7)
Adim△7 = A diminished triad with a major seventh
Further to Aaron's answer: the 'flat 7' found in a dominant seventh chord is there due to the chord being made up from notes diatonic to a different key.
As in C7 - C E G B♭. Those notes are diatonic to key F, rather than key C, which is also where it's found.
And 'augmented seventh' is a bit of a misnomer! There's a seventh, and the fifth is augmented. There's no augmented seventh interval in sight. If that interval was there, in, say, key C, it would incorporate a B♯, sounding rather like the root note, C!
There's a logical reason for why this is the case:
That's because an augmented 7th chord has nothing to do with an interval of an augmented seventh! In fact, the full name for this type of chord is Augmented Dominant 7th.
C E G# Bb
(These are all equivalent ways to write C augmented 7th. You may also see parentheses involved, which indicate that the notes inside are to be explicitly added to whatever remains outside of the chord. In this case, grouping makes no difference at all since 7th chords are already defined as triads with added 7ths.)
Now it should be easier to split up that +7 symbol into the + and the 7. The + represents an augmented triad, and the 7 indicates that the chord is of dominant 7th quality. If it helps any, think of the chord as a dominant 7th chord first, then augment the 5th afterwards. That should help you make the association between the interval pattern and the symbol (and it's good practice for the many altered dominant chords that act the same way)!
For fun: there's also a chord called Augmented Major 7th, labelled +maj7. This chord is an augmented triad with a major 7th (or, potentially more helpfully, a major 7th chord with an augmented fifth). Not a very common one in tonal harmonies, but a good one to know about!
We sometimes call the 'C7' chord type a 'Dominant Seventh'. Because its most common occurrence is as the dominant chord in a major or minor key.
So C7 doesn't have a 'flattened 7th'. As the dominant chord in F major it has a normal, diatonic 7th.
Most chords with '7th' in their name are variations of this basic 'dominant 7th' shape. 'Caug7' is C7 with a raised ('augmented') 5th. The 'augmented' refers to the 5th, not to the 7th. 'C7(b5)' is C7 with a lowered 5th.
But in 'Cdim7' the 'dim' DOES refer to the 7th (as well as to the 5th). Once again, music theory has frustrated our desire for a nice, neat set of rules.
"'7th' in a chord name means it's based on the dominant 7th shape, except when it doesn't."