The string theory model is structured from the start to reproduce existing theories as a consequence, so the problem here is a lack of understanding on your part rather than a mistake in the theory. I do suspect that most people who are working on this stuff (or any stuff for that matter) don't think very hard about the basic phenomenological claims to the point where they can explain them well.
In particular your model of electric fields isn't very good. An electric field's flux around a volume reveals the presence of a particle in that volume. That's not because the field, a bunch of vectors in space, "happens" to integrate to something nonzero if there's a particle in that volume: it's because in some sense the presence of a field with a nonzero flux and the presence of a charged particle are the same thing; the particle is the existence of a divergence in the field within that volume.
Moreover in QFT (and this part is handwavey as I only learned enough to vaguely understand this, but it's better than nothing) the presence of the "field" ends up looking like the sum of what you get if you integrate over every possible way of a emitting or receiving a photon at that point; the accumulated integrals destructively interfere in such a way as to produce a value which reflects where the particles are. So very roughly idea of a field existing at a point and having a "value" is like saying: there are a bunch of things out there that I (a charged particle) can detect by exchanging photons, and the accumulated effect, when you consider all the different quantum superpositions of ways of doing that, is a single vector which induces a force on me. Other fields add up to more complicated objects than vectors.
Once you look at things like that, there should be no objection to how strings and electrons might interact. Whatever's going on at that string level averages out over larger timespans to just look like electron field. Not dissimilar from how all the individual charges in an atom average out to look like a single charge (but are perhaps detectable if you get up really close, in dipole and higher moments, or in how the atom deforms / reacts to nearby charges).
It might help to be aware of the concept of a topological defect (https://en.wikipedia.org/wiki/Topological_defect; there's a great explainer article somewhere that I can't seem to find) as a reductive picture of what a particle "is". I've heard this doesn't work for fermions for some technical reason, but in any case it's very useful as an illustration of the sort of thing that a particle "can be": a vortex in a material can act like a particle and even exhibit attractive/repulsive forces. So I picture the string theory model as answering the question: what kind of substrate could produce vortexes and other "defects" that act like the particles we see? Dunno if that's accurate but it seems like a natural question to me, anyway.