Why are some ions mutagenic and others are not?
You are confused about the cause of mutagenesis in alpha particles.

The reason why an alpha particle is mutagenic is not that it is an ion, it is that it is an ion carrying a great deal of kinetic energy. Alpha particles are produced in energetic nuclear reactions and carry a great deal of energy away from the nucleus that emits them (a typical alpha particle has about 5MeV of energy, enough to break a typical chemical bond). It is the kinetic energy that matters not what the ion consists of. Heavier ions with that sort of energy would be just as bad for biological systems as are other forms of energetic radiation (beta particles are just electrons but will still do damage, gamma rays are just powerful photons but will dump enough energy into molecules to break DNA).

Non-energetic helium nuclei would pick up electrons extremely rapidly from the first molecules they encountered (probably from air even before reaching your skin) and would probably be as harmless to you as the helium filling a balloon).