Some history
Is it because three more quarks $(c,b,t)$ were not discovered at that
time?
People even didn't know about quarks when Gell-Mann was discovering his "eightfold way" (and long-long time after that). They only knew about pions, nucleons and other strongly interacting particles, and characterize them by the spin, electric charge, isospin. The number of these particles was very large, and people wanted to describe this strongly-interacting zoo in a simple way.
What was the motivation for implementing a $SU_{f}(3)$ flavour
symmetry
There were underlying reasons for this.
One of them was $\sigma$-model constructed by Gell-Mann and Levy in 1960 in order to describe nucleon-meson interactions. It explains many phenomenological things like vector current conservation, PCAC and Goldberger-Treiman relation. The fundament of $\sigma$-model is approximate $G \simeq SU_{L}(2)\times SU_{R}(2)$ global symmetry group spontaneously broken down to $SU(2)$. This $SU(2)$ group is called the isospin group. The isospin symmetry was later extended on all strongly interacting particles, since it was realized that the strong interaction (approximately) respects it; other particles formed isospin multiplets.
Another reason was observation of some strongly interacting very massive (in compare to pions) particles which have relatively long life-time (even if their masses are large): precisely, they are decayed through weak interactions, which characteristic life-time is much larger than the strong interactions characteristic time. Gell-Mann proposed to explain this phenomenon by introducing new quantum number called strangeness; the strangeness is preserved by the strong interaction but is violated by the weak interaction.
And another important reason was the Gell-Mann-Nishijima formula
$$
Q = I_{3}+ (B+S)/2,
$$
which relates the electric charge of stringly interacting particle to the linear combination of the isoapin projection $I_{3}$, the baryon number $B$ and strangeness $S$. This phenomenological expression describes hundreds particles!
So, to conclude: at the time of implementing the flavor symmetry all strongly interacting particles were characterized by their spin, isospin (and isospin projection), strangeness and electric charge. There was not enough for describing them in a simple but systematic way. But above reasons signaled that the true (approximate) global symmetry group of the strong interaction is some group larger than the isospin group $SU(2)$.
Gell-Mann then proposed to extend the global symmetry group to $SU(3)$. This group is what we know called the strong interaction flavor symmetry group. In his approach, different strongly interacting particles form multiplets with definite spin and similar masses. These multiplets are contained in the tensor product of (some) fundamental representations $3, \bar{3}$. Gell-Mann identified, for example, the pseudo-scalar mesons as the octet contained in the $3\otimes \bar{3} = 8 \oplus 1$ product, while the nucleons in the octet contained in the $3\otimes 3 \otimes 3 = 10 \oplus 8 \oplus 8 \oplus 1$ product.
What was the triumph of the symmetry (1)
So, the triumph of this approach was that it provided systematical way to describe all strongly interacting particles by representing them in a very simple way based on a universal property of strong interactions - the flavor symmetry. It also predicted the new particle called $\Omega^{-}$ and relations of masses of particles from given multiplet. But from the modern perspective it was the main theoretical reason to introduce quarks. It also explains the Gell-Mann-Nishijima formula, since three quarks $(u,d,s)$ can have only three independent quntum numbers, and hence if there are $m$ extra quantum numbers, then there must be $m$ relations between all $3+m$ quantum numbers leaving only 3 numbers independent.
The modern view
From the modern point of view (the QCD one), these fundamental representations $3, \bar{3}$ are physically recognized as quarks triplets $(u,d,s)$, $(\bar{u}, \bar{d},\bar{s})$. The underlying global symmetry of the QCD is $SU_{L}(3)\times SU_{R}(3)$ (here I "neglected" the $U_{A}(1)\times U_{B}(1)$ group, which is not relevant here). It is approximate symmetry, since quarks have relatively small masses breaking it explicitly. This symmetry is spontaneously broken down to $SU_{V}(3)$. The $SU_{V}(3)$ is what You can call the flavor symmetry. The fact that mesons (for example) are contained in $3\otimes \bar{3}$ means that they consist of two quarks (some quark and some anti-quark); the nucleons are contained in $3\otimes 3 \otimes 3$, and this means that they consist of 3 quarks.
and why did it work when the Standard model shows that the symmetry is
something else as given in (2)?
Of course, there are three other quarks - $c,t,b$, and formally You may think about $SU(6)$ symmetry. But their masses are so large (more than $ \text{ GeV}$) that they don't make the contribution in the observed particles spectrum: the $SU(6)$ symmetry isn't even approximate symmetry at most interesting scales. Even $SU_{f}(3)$ symmetry works bad below $100 \text{MeV}$ because of large mass of $s$-quark.