are massive bosons in fact fermions?

Well, yes. It's quantum field theory, not quantum point-particle theory. The electron is field. It isn't something that's "got" a field.

In QFT the electron is said to be "an excitation of the electron field". I don't think that says enough as it happens, but nevermind. The point is this: the electron is just field.

You have explained what QFT is. In QFT, there is a fermionic field whose excitation the electron is said to be and there is also such a thing as a bosonic field, and similar reasoning applies to it: an electro-magnetic field is not something photons have, it's something they are.

There is just a little problem: the electron seems to have a field: the electric field
The electric field is not yet quantized. What is the quantum of the electric field?
 
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