The SHG process massively drops your power output. Orders of magnitude of power loss. Which for a laser pointer is no big deal, they are already really low energy, and in the end just need to be visible. But for a laser cutter, you are talking about moving your GlowForge 40W laser down to somewhere around a 40 mW laser (I don’t know any simple formula to calculate exactly what the losses are, but a 0.1% yield is a reasonable estimate). At that power level, popping a balloon is only barely possible.
The nonlinear optics devices let in the laser (photons), and then bounce those around atom to atom in a matrix for a little while (in our terms, no time at all. In light terms, trapped). A new set of photons comes in, and also get trapped, into the same atoms in some cases, and this results in higher energy photon release happening.
This doesn’t happen for absolutely every photon released. Many are still released at the same energy they came in at, and some are released at intermediate energy levels (the levels are quantized, so having intermediates depends on energy in, and material of the optics). If your use case demands only having the higher frequency laser in the beam, you need to pass through a filter to trim off the non-doubled portion of the beam.