Nice question. I hope that someone more knowledgeable than me will step in
and bring more information, but let me share what I think.
It doesn't seem to me that Grothendieck had an idea of how to prove this fact algebraically.
It is true that Grothendieck had many ideas, and that some of them was so ambitiously genial that they would take year even to him to be realized, the most notable example being the
idea of the proof of the Weil's conjecture. But in this case, he would say that he had the idea. For example he announced his strategy for the Weil conjectures very early, at the ICM
1958, and in the introduction of the EGA. From my readings of Récoltes et Semailles,
I believe he would have considered unethical to raise a question without saying everything he
had in mind on the subject.
This being said, back on the subject of determining algebraically the étale fundamental group
of the projective line minus n points, here is what he says in SGA 1 (remarks 2.7 page 267):
"A cet égard, la situation de la droite rationnelle privée de $n$ points, et l’étude des revêtements d’icelle modérément ramifiés en ces points, est plus sympathique, puisque la considération des groupes de ramification en ces $n$ points fournit $n$ éléments du groupe fondamental à étudier, dont on montre en effet qu’ils engendrent topologiquement ce groupe fondamental, comme nous verrons ultérieurement. Mais même dans ce cas particulièrement concret, il ne semble pas exister de démonstration purement algébrique. Une telle
démonstration serait évidemment extrêmement intéressante."
So he says that one can algebraically construct $n$ elements in the $\pi^1$, and
using transcendental methods prove that those elements topologically generates the $\pi^1$.
But he doesn't seem to imply that he has the slightest idea about a purely algebraic proof.
Edit: An other argument came to my mind. The subject of the étale fundamental group of the
projective line minus n points is one of the very few question that interested Grothendieck a lot in the two part of his mathematical life (before his abrupt depart from IHES in 1970,
and after that). So if he had an idea about how to prove this fact around 1960, and not the time to follow it, would he not have come back to it later when in the nineties if he was interested in the "Dessins d'enfants" and all those anabelian things. To my knowledge he didn't. (Again this is just a guess, that would happily yield to facts or even more educated guesses)
