I'd be surprised if many p2p players have a "complete collection." I pay some, and I'm nowhere close to a complete collection, nor will I ever get there. And I'm more than fine with that. There are a lot of crappy epics and legendaries in previous expansions that are a complete waste of dust. I'm happy I never opened them.
A better question is how much do you have to play (since that's the only way for f2p to get the gold they need for the card packs) to be able to build the decks you want to play? And the answer to that is: it depends on how many decks you want to build and how expensive they are.
As a F2P player completing my dailies and then playing a few extra games on ladder over the 4-month DoD expansion, I collected enough gold and dust for 9/25 AoO legendaries and many, but not all, of the set's epics. (One of these AoO legendaries came from a bundle I bought during Scholomance Academy.)
A similar approach to playing and purchase of the SA pre-release mega bundle and Felosophy and Horseman Uther bundles netted me 18/25 SA legendaries and a full set of other rarity cards.
Playing since day one. Never missed a card back, bi monthly legendary, have dropped money on expansions....
I only completed the standard set through natural attrition last year 2020. And average about 90% of standard releases by the end of an expansion. I only purchases pre releases and not always, I have never bought raw packs or mini releases, I have o ly used gold for those.
This is just my experience but I can say I have above average pack openings (1/15 legendary packs since I've been tracking, 2016) so my collection should be more "complete" than most.
Point being, I still don't have a total complete collection and have only ever had 1 total golden deck, which was purpose built since I had the dust for it.
So no, a F2P player will affective never achieve a complete collection, not withstanding playing 247365