I feel like this answer with only restate you know and is in the book, but...
If consonance and dissonance as stated in the book are considered synonymous with stable and unstable, I think the concept is clear.
Root position chords are stable and synonymous with consonant.
Inverted chords are unstable and synonymous with dissonant.
That all stated pretty clearly in chapter VIII, item 2.
The part that follows about a sixth as a suspension resolving to a fifth does seem odd. The point would seem to regard inverted chords as not bona fide chords...
...so that second chord is not
V with a suspension.
I did a quick scan of the chapters before this. It looks like it teaches a rule of the octave kind of harmony. It shows how to harmonize a bass line where the harmonies are predominantly tonic and dominant chords only.
If you want to analyze harmony in terms of tonic and dominant, then the chord of the sixth example above - where the sixth is considered a suspension - is not really regarded as an inverted mediant
iii6 but as a dominant
V with a suspension.
That's my quick impression. Call the sixth a suspension so the chord can be interpreted as a dominant.
Some questions stick in my mind, like this one.
I stumbled upon this today...
...from Kostka & Payne, Tonal Harmony, the still call the sixth a consonance, but I think it's clear how this connects with Ouseley and regarding the tone forming the sixth is a non-chord tone and the proper chord isn't the inverted chord of the sixth.
Also, I was playing this sequence drill from Fenaroli's "rules"...
...if you look at the first three half notes in the bass, the chords are
G: I V6 IV6. That means the
E5 in the soprano has a dual role, an escape tone from the initial tonic, but a suspension on
V6. This is from a book about figured bass so it contains no chord analysis. But, I think it is clear there is no
vi6 chord in the first bar. It isn't explicitly called a "dissonant sixth" but it has the feel of an escape tone, a non chord tone.
So, most non chord tones are dissonant. But some non chord tones of the sixth aren't technically dissonant, but their not fitting the chord lends them a kind of tension similar to dissonance. Ouseley calls this a dissonance of the sixth. It's "dissonant" simply because it isn't a member of a root position chord.