In Quantum Electrodynamics, things are simple, because the photons are uncharged, so they themselves do not interact through Electromagnetdism, . Of course, there are still divergencies, and you still do need to renormalise, but things are very simple, compared to...
Quantum Chromodynamics,. Things now get much more complicated. Well, the Lagrangian Density takes almost the same form as in Quantum Electrodynamics, but compute anything, is a horror. Why? Gluons themselves have colour charge. SSo the additional gluon potentials, which ignoring constants, is $A^\mu=\nabla^\mu-\partial^\mu$ in the contra - variant form, have colour charges themselves, and interact through the strong force themselves. And thus, there is a lot of problems with the divergencies, but still, it is renormalisable.
Since everything was renormalisable, strong coupling, weak coupling, and they could make a TOEEG (theory of everything except gravity), called the standard model, they thought the same elegance could be extended to gravity, to general relativity. I mean, the standard model does incorporate special relativity, so why not general?
The most obvious way to do so, was to introduce a gravitational quantum, the graviton. But sadly, it was done in a very naive way. The gravitons themselves contributed to the gravitational field, just like in Quantum Chromodynamics, and whatever it is, the end result diverged. But unlike in Quantum Chromodynamics, for Quantum Gravity, it was non-renormalisable,.
So, a less naive theory of gravitons is required, and such a theory is string theory.
So, thus, it was not about the standard model not sensing a requirement to include gravity, it couldn't. One needs string theory for that; to make everything known to man, and everything that is true, to come out naturally.
There are, of course, other TOQGs (theories of quantum gravity), but the problem is that most of them are lorentz asymmetric (cf. Lubos Motl's criticism to Loop Quantum Gravity), they don't allow for other forces (interactions) other than gravity, they are mathematically inconsistent, etc. Or they just lead to string theory (e.g. supergravity, kaluza - klein theory).