I have wondered for a long time why the wavefunction associated with a photon is a tangible and measurable change in an electromagnetic field whereas (all?) other particles have a less tangible probability amplitude associated with them.

The photon wavefunction isn't quite the sinusoidal electromagnetic field variation, but the latter is very real and very detectable:

Imagine a photon is travelling from A to B. You can measure this sinusoidal field variation at some point along the path. Now step away from the path, and measure it again. You will still measure a field variation, but it will be less intense. Step further away and it's even less intense. Integrate over a volume and say to yourself that the photon is a "pulse" of electromagnetic field variation, and you perhaps get a sense of "many paths". The photon, which might have a wavelength of say 1500m, does not exist as a billiard particle at any point location, instead it's a wave function distributed through space. It is however emitted or absorbed as a unit, so it can look very small. I don't see anything remarkable about this, after all i can shake the whole of a rubber mat using only my hand.

An example is an electron which exhibits interference and diffraction but for which the wave function appears as just an abstract probability amplitude.

IMHO the best way to conceptualize an electron is as a photon that is "wrapped" via pair production into a tight twisted circular path, something like a moebius strip. Now instead of being a transient passing field

*variation*, it's a permanent standing field. The electromagnetic field is thus a part of what the electron is, and part of its wavefunction. You might say that the centre of this is at some given location, but again you can't say that the electron is a point particle at that location emanating a field. The electron itself is a "quantum field". Unfortunately IMHO some educational material depict it as something very small and pointlike, much smaller even than its Compton wavelength.

Are there any other particles for which the wavefunction is a sinusoidal change in a field?

Not to my knowledge.

If not, can any insight be drawn from this?

I think so, see

Topological quantum field theory wherein the quantum field for say a photon has a different topology to that for an electron. It's related to knot theory, and in a nutshell advocates that the electron is essentially a 511 keV photon that's "tied in a knot". However TQFT doesn't seem too well-known these days, even though people like Witten and Atiyah have been involved in it. Some would argue that it isn't mainstream, but I think it will be.

If a Higgs Boson is found, would its associated wavefunction be tangible gravity waves?

No, because a gravitational wave is actually rather like a photon. I suppose you could call it a particle, particularly if you could keep pace with it and see it as a "bump" rather than just a wave. It's a particle in the same fashion as a single-photon electromagnetic wave, but the E=hf quantum aspect doesn't hold, so it's arguably less particle-like in nature.

As the photon mediates the em force, the graviton mediates the force of gravity and the pion the nuclear force (I think this is right but it is from memory), do these particles have tangible waves in their associated fields that produce identical results to a probability amplitude?

I'm afraid the graviton is speculative. It isn't part of the standard model. And whilst the gluon is said to mediate the strong force, we don't actually see free gluons, just as we don't see free quarks. So we can't talk about tangible waves like we can for photons.