If you were to find a gym in a city that trains like this, what exactly would you type in in google? Core builing?

A search for 'Core gym' would probably lead to the website in my link, as that is what the gym is called. For gyms in your area, you should look for 'functional strength training' or any gym that supports or offers 'crossfit' training. They are still few and far between sadly. Far more people doing this sort of stuff in garages and on parkland, than in gym facilites.
Don't let the door hit ya' where the good lord split ya'