Hello,

The correct work-flow is like the first scenario:

1)install June theme -> Activate;
2)install June-child;
3)import demo;
4)activate Child theme;

Theme Customizer is in fact the WordPress’s default customizer, but developed to add to it more options in order to fully customize every theme element and have as many settings to change theme style as possible.
So to have it open faster you only need to give it more memory limit or just have a good/fast computer.
It has no loading issues if opened from a Mac computer. I know this is not a requirement but having better hardware does effect all the programs performance as well.
I can suggest you to open the Customizer from Pages options (go to page option > Customize Page & Options) instead of Appearance > Customize. It’s faster that way.

Best regards!