One problem is alignment as described, which is extremely finicky on most such greens.
Another problem is wavelength match: The pump diodes are usually described as '808 nm' but that wavelength is not always exactly spot on. It has to be exactly right to match the crystal in the laser to get the best results.
In these small laser pointers you have no control over the temperature of the laser diode or either crystal, making it a bit of a gamble if it matches up or not. It's fun to experiment with for sure, but don't get your hopes up too high. You will probably get some green light out of it, but how much is anyones guess.
Make sure to wear proper eye protection against IR though, there is a LOT of power in those pump diodes!
Laser201698, were you able to remove the diode? Was it without a can? Any optics on it?
To put everything together get a thermal epoxy. First practice without it and with a reduced laser diode current.
If you have some tools at home to move precisely the crystal head it would make life simpler. Something like - you placed a crystal in a mill collect and hold a laser diode part in a vice, then just program your CNC machine to do a work for you LOL