Set $f_t(z)=\phi(t,z)$.
Notice that entire functions $z\mapsto f_t(z)$ all commute with each other.
I. N. Baker proved that if $f$ is an entire function with a repelling fixed point
then the set of entire functions with commute with it is countable.
MR0147650
Baker, Irvine Noel
Permutable entire functions.
Math. Z. 79 1962 243–249.
On the other hand, in a later paper he proved that for every non-affine entire function some iterate has a repelling fixed point.
MR0226009
Baker, I. N.
Repulsive fixpoints of entire functions.
Math. Z. 104 1968 252–256.
Combination of these two theorems settles your question. Indeed, start with $f_1$. Then $f_n$ with some positive integer $n$ has a repelling fixed point. Then the set of entire functions which commute with $f_n$ is at most countable, contradiction.
Once we know that all $f_t$ are affine with respect to $z$, the rest is easy: commutative sub-semigroups of the affine group are just what you listed.
EDIT. Second theorem of Baker was based on the Ahlfors Islands Theorem, which was considered "heavy machinery" at that time. Since then both Baker's proof and the Ahlfors Islands theorem were very much simplified. For an elementary proof of the Baker theorem see, for example,
MR1782673 Berteloot, François; Duval, Julien Une démonstration directe de la densité des cycles répulsifs dans l'ensemble de Julia. Complex analysis and geometry (Paris, 1997), 221–222, Progr. Math., 188, Birkhäuser, Basel, 2000.