The effect of a spacetime transformation isn't just to redefine straight lines along which particles move. It means measurements (e.g. lengths, areas, time intervals) are different depending on where you are in the spacetime. The are forces don't come with these "extra" effects - an EM field doesn't stretch and contract space.
However, there are a lot of parallels between electromagnetism and relativity! Quite often relativity effects are introduced with an EM analogy, e.g. gravitational waves (which have polarisation) and electromagnetic waves ie photons (which also have polarisation). Note though it really is an analogy - they are fundamentally different things in both reality and mathematical form.
The main difference though is that gravitation (probably) doesn't have a mass of its own, while EM fields do. Plus, matter does not react to EM fields in the way it does towards gravitation. I.e. light is not "pulled" by EM fields. However, these are technicalities. If all matter reacted to EM fields the same way it would to gravitation, would that make EM fields no force either? Or put another way, gravitation act universally on all particles, while EM fields do not. That necessarily has consequences when it comes to relativity. However it seems odd to argue that general relativity would exclude gravitation from being a force. If it acted only on a subset of particles, it would likely be in the same position as EM fields, and suddenly become a force again?
> The main difference though is that gravitation (probably) doesn't have a mass of its own, while EM fields do.
Both gravity and EM fields have energy which is what couples to the gravitational field. Neither of the fields has mass, though.
> If all matter reacted to EM fields the same way it would to gravitation, would that make EM fields no force either? Or put another way, gravitation act universally on all particles, while EM fields do not. That necessarily has consequences when it comes to relativity. However it seems odd to argue that general relativity would exclude gravitation from being a force. If it acted only on a subset of particles, it would likely be in the same position as EM fields, and suddenly become a force again?
This is very well thought. Indeed, the equivalence principle, the fact that gravity couples to everything in exactly the same way (and that includes gravity itself as per the previous clarification) lurks behind our ability to reinterpret gravity in a geometric fashion. After all, if something didn't interact with gravity in the same way as everything else we could establish an experiment to differentiate if a spaceship is accelerating or stationary under a gravitational field (see Einstein's mental experiment) by measuring how that thing behaves. And that same fact would stop us from interpreting gravity as curvature of spacetime itself.
To your last point, speaking of forces is probably antiquated anyway, although still in use partly for historical reasons partly abuse of terminology. Preferably we should use the term "interactions", after all some of the "forces" do not result in push or pull as we usually understand a force in Newtonian mechanics but in things like color change. Others, like the gravitational "force" can be expressed entirely as spacetime geometry. But discussing semantics is quite pointless so as long as everyone understands in what way the term "force" is an abuse of terminology it's OK to keep using it.
Also a layperson here. Can you give an example of how we can tell that EM fields don't "stretch" space, but gravity does? Is it just about how light behaves in those fields or is there something more to it?
As mentioned below, one way to think about it is that EM only affects charged particles (and depends on their charge), whereas because gravity is acting on the underlying space-time it has a universal effect on everything (including light).
We can pretty much boil EM down to: like charges repel, unlike attract, strength is charge1*charge2/distance^2. What about magnetic field, photons, QFT etc?? None of this exhibits effects which could be described as stretching space-time either.
But we cannot do the same with gravity. An explanation like the above but for gravity (which is traditional Newtonian) leaves out many, now observed effects such as:
-> time dilation (GPS relies on this calculation) (measures time stretching and contracting)
-> gravitational waves (LIGO) (measures space stretching and contracting)
How do we _know_ any of this? People propose theories, those theories are then tested against experiment. AFAIK to date there is no experimental evidence suggesting EM stretches space, and no theory proposed that includes such an effect and correctly matches experimental data. That's the most holistic answer (but unfortunately one you just have to believe unless you have a lot of spare time!)
However, there are a lot of parallels between electromagnetism and relativity! Quite often relativity effects are introduced with an EM analogy, e.g. gravitational waves (which have polarisation) and electromagnetic waves ie photons (which also have polarisation). Note though it really is an analogy - they are fundamentally different things in both reality and mathematical form.