wolram,
one way to straighten it out in one's mind is like this:
in order to make an electron cease to exist, and just have light left over, nothing but a flash of light, you DO need an anti-electron to cancel it.
so in a sense you are right. same with a proton, if you want to have nothing left but a flash of light----if you want to get rid of all the massive particles, all the "matter", and have nothing but what is obviously energy, a flash of light, then you DO need an anti-proton or some equivalent kit of anti-particles to react with it.
but a larger object like an atom or a cannonball is made up of LOTS of these particles, lots of protons and electrons (and neutrons...)
and THE MASS OF THE BALL DOESNT JUST DEPEND ON HOW MANY of each kind there are but rather on how they are arranged and WHAT THEY ARE DOING.
so if you take a cannonball and change what the particles in it are doing you can change the MASS
or if you rearrange the particles in the iron atoms into some other kinds of atoms that also will sometimes change the mass, it is an empirical fact backed up by experiment
that doesn't involve any antimatter and it tends not to change the NUMBERS of various type things----you just put in some energy and change how things are arranged and what they are doing and that changes the mass---according to the formula.
or the things spontaneously rearrange in the cannonball and some energy comes out and now it has less mass---according to the formula
(but no antiparticles were involved because the numbers of different types didnt change)
I find that real intuitive and easy to accept. How about you, wolram?