It's literal. If psychology.stackexchange.com existed, they could help better than me, but basically, if Harry opens his mouth words may come out that indicate his true response to Dudley's appearance.
These words would very likely not include handsome, grown up and pride inspiring.
Given the context of how the mentioned protagonists already feel about Harry, he decides it is safest not to say anything at all. He may start off trying to say words that the other members of the household want to hear, but during any ensuing conversation, he will have to say something reflexively, and this might be closer to what he actually thinks. The expected response from the household to that would not be likely to be pleasant for Harry, so he literally is not trusting himself to speak.
Additionally, he has concerns about breaking into uncontrollable laughter. Devoting effort to talking would make it more difficult to prevent this from happening. I would have picked this as the reason, but in that case a better lead in sentence would be 'Harry didn't trust himself to open his mouth.', so I feel the speaking is more related to the fact that Vernon and Petunia have both uttered highly positive sentences and Harry can't carry that off. Contrast this with either of the twin Weasely brothers, for instance, who would have no problems in trusting themselves to speak.
Harry's mum was a good witch and his dad was a good wizard.
Yer mum and dad were a good witch an’ wizard.
How good were they? They were as good as any witch or wizard I ever knew.
Yer mum and dad were as good a witch an’ wizard as I ever knew.
These constructions are maybe, possibly, just a little bit too colloquial for, say, an exceptionally dry academic journal; but anywhere else they’re fine.
Best Answer
In some very limited contexts, yes. As the example you linked to ("The one, the only...") demonstrates, the word "only" can be used to mean "best" in a somewhat indirect way by saying that some person, entity, etc. stands alone in their field, implying that they are the first ever to achieve such a high level of expertise or skill, and thus they are, by extension, the best in their field.
However, in the above context (Harry Potter), no. The word "only" means that the folks referred to by "some o' the best" were literally the only ones in their family lines to have magic abilities; no other person before or beside them had such abilities.
This doesn't imply that they were the best magicians. In fact, quite to the contrary, it seems that some other character (the "he" in "what does he know") has implied that Harry's lack of magical heritage is in fact a very bad thing, not a good thing. (If I'm interpreting this correctly; I know little to nothing of the story.)
However, the speaker does state outright that the characters he refers to are "some o' the best [they] ever saw". So while "the only" doesn't mean "the best", "the best" does mean "the best".