If all particles are fields, why does first quantization work for some particles?

First quantisation cannot accommodate pair-production, which is non-negligible at energies comparable to the mass of the particle. Therefore,

First quantisation only works for massive particles, and only in the range of energies that make motion non-relativistic.

Of course, this doesn't mean that a massive particle can always be modelled in a first quantisation scheme: for example, another requirement is that the particle is stable, and unaffected by confinement, etc.

Towards the edit, let me stress that the localisation of particles in a relativistic context is a very subtle issue (cf. this and this PSE questions). A nice reference for this problem is No place for particles in relativistic quantum theories?, by H. Halvorson and R. Clifton.

AccidentalFourierTransform gives you physical reasons but first quantization also came about partially for both historical and pedagogical reasons.

If you take the electron, a hundred years ago it was cutting edge science to match up the predictions of the "weird" quantum world to what was "obvious" in the classical world.

In something similiar to the simple (but misleading) Bohr semi-classical picture of the atom, first quantization makes certain simplifying assumptions and mixes classical and quantum descriptions.

In first quantization, some physical properties, such as electric or magnetic fields, and the potential wells associated with them, are treated classically, but their effects are seen as changes in the quantum description (utilising wave functions and matrices) of particles, especially the electron.

Actually, imo there is no other real alternative to learning QM basics, except by treating quantum systems as they are done in the first quantization method. As John says in the comments, otherwise you would need to know more about QED and QFT. If you were treating a system of say, an electron in a magnetic field without the use of first quantization, it would be a confusing chicken and egg like affair, with the need to explain the $E $ and $B $ fields in field terms.

Its much easier to learn the basics of QM this way, and allows for the particle's motion and spin to be dealt with in the first "explanation" of how a particle behaves quantum mechanically and without the distraction of having to explain non essential "external" effects.

Second quantization then appears after you upgrade from the Schroedinger equation to the Dirac equation, with it's built in description of spin, particle production and the later introduction of the field concept as the basis for modern particle physics.