You can indeed, I started off my collection this way.
Just don’t make the mistake I made and buy any wild packs or split your purchases across too many sets. You’d be surprised how little you get from 60 packs. Focus on the current Standard sets with the best cards and craft the rest. I’d also wait for and take advantage of all Bundle deals and pre-orders rather than buying packs at normal price.
Alternatively, you can reroll a quest you don’t have the cards to complete and continue to reroll said quest until you get one that you can complete, while completing the other quests you get.
Also, whether you believe it or not, there are videos of people creating their free to play accounts, right there, on the video, then playing through the tutorial missions and playing through the practice matches to unlock the cards then collecting their bonuses etc… Your belief in these things is not necessary or desired, they happened, whether you believe it or not. Your belief, or disbelief, cannot withstand the actual evidence, the proof.
Lastly, what daily grind do you speak of? It takes longer to run an Arena run than it takes to complete a quest and win an extra 3 to 6 games. And completing a quest and winning 3 to 6 games has a predictable and guaranteed outcome.
Where, with Arena, no matter your skill level, your reward outcome is NEVER guaranteed, because Arena, by its very nature, is a gamble.
And in addition to that, where does the gold you used to start an arena run come from? How does a free to play player learn how to play arena well if they have no gold from completing quests? And in the process of learning Arena, there is much gold lost.