Does current fundamentally determine the speed or the quantity of electrons in a curcit?
Assume you have a conductor of volume \$A l\$ (cross-sectional area = \$A\$ and length = \$l\$)

Let \$n\$ be the number of free electrons per unit volume of the conductor, then the total number of free electrons in the conductor is given by \$Aln\$.

If \$e\$ is the charge, then net charge \$Q\$ is, \$Q = A l n e\$

Let \$v\$ be the velocity of electrons due to this potential difference. This is called as drift velocity of the electrons.

Let \$t\$ be the time taken by an electron to cross the length of the conductor.
Then, \$t = l/v\$.

As current, \$i = q/t\$, we get \$i = A l n e/t\$. On substituting \$t\$ from above equation, we get \$i = A n e v\$.

So current is directly proportional to drift velocity of the electrons

But this drift velocity is in terms of few millimeters per second. It is because of the large value of \$n\$ (no. of electrons per unit volume) that effect of velocity gets magnified.