Call a group terminally n-generated if it is a subgroup of an n-generated group.
For example, even though $C_2 \times C_2 \times C_2$ is 3-generated, it is terminally 2-generated because the group with presentation $\langle s,t\mid t^6, s^2, t^3st^3s, t^2st^2st^2s\rangle $ has it as a subgroup, with its three generators being $a = t^3, b = s, c = t^2st^4$.
Because symmetric groups are 2-generated, every group is terminally 2-generated as it is a subgroup of a symmetric group.
Are there any 2-generated groups that are terminally 1-generated?
A $1$-generated group is cyclic; a subgroup of a cyclic group is cyclic. If by "$n$-generated" you mean "can be generated by $n$ elements, but cannot be generated by strictly fewer than $n$ elements", then the answer is "no", because if $G$ can be embedded into a cyclic group, then $G$ is cyclic and hence either $1$-generated or $0$-generated (if trivial).
On the other hand, one often says that a group is $n$-generated by it can be generated by $n$ elements, but may be generated by fewer elements; in that case, every nontrivial cyclic group is $2$-generated (you can always through in the identity, or some other non-trivial element in addition to the generator). In that case, the $2$-generated groups that can be embedded into a $1$-generated group are precisely the nontrivial cyclic groups.
You don't tag this as being about finite groups, and don't specify finite. So note that your argument about symmetric groups only works for finite groups${}^*$. However, it is a classic theorem of B.H. Neumann, Hanna Neumann, and Graham Higman that every countable group can be embedded into a $2$-generator group, so that would show the result holds for all countable groups. The result cannot hold for uncountable groups, since a finitely generated group is necessarily countable and hence all its subgroups are countable.
${}^*$ It isn't even always clear what an "infinite symmetric group" is; sometimes it means the group of all bijections from the set to itself, and sometimes it is the group of permutation with finite support (so for example, the "symmetric group on $\mathbb{N}$" would be the direct limit of the finite symmetric groups $S_n$). Note that the set of all bijections from $\mathbb{N}$ to $\mathbb{N}$ is uncountable, so it cannot be $2$-generated.