Don B
No, I don't think you (one

) need/s to get the references, although I also think that you do need to get the references. Or, rather, that not getting the references confirms Pope's nightmares.
Clearly, there is a disjunction between the easy familiarity with Pope's references that Pope's audience possessed* and the situation for most of us - and, certainly, me until I 'did' Pope at UCL as an undergraduate.
And that disjunction is one that began to widen, I think, within possible audiences for Pope's writing in the C18. The whole issue, in Pope, of a mass readership & cultural 'collapse'.
So, complexly,
The Dunciad plays both with reference recognition & reference blindness. The satire is both the bathos of the references applied to their targets (as in
The Rape of the Lock, which I agree is a perfect allegorical machine. Quite breathtaking) and the reality that those references would be (blissfully) lost on the targets of the satire. So cultural loss (whether, unambiguously a loss is another matter, since it allowed for other discourses. Or, rather, whether it was a loss in the way that Pope & his circle would have conceived it. But it is anger at trivialisation & ignorance).
* whatever Pope's audience was. I'd guess it was a variable audience, variable according to the poems, and there is clearly an element of in-jokiness and a select circle that operates in Pope, runs beneath or across or ... the public 'syntax' of certain of the poems.
What I was driving at, I think

, was that within Pope's writing the references are loved, internalised, and that gives the poems extraordinary rhetorical force. In a way, what a sort of hypostasised Nature can, at times, be for Wordsworth, Classical Culture is for Pope. (Clearly, both are much more complex poets than that: but when Blake wrote that reading Wordsworth had given him a stomach ache which almost killed hm, he was offering a somewhat Pope-ian critique of a type of spiritual materialism).
So, there's no doubt in my mind that that rhetorical force comes across without reference familiarity. But I doubt that Pope would have found that a happy thought. If that makes sense.