"Why don't Republicans support an increased minimum wage?"
MOST Republicans don't support it, and the main reason they do not is because the minimum wage concept is foolish economics. Creating a minimum wage law and periodically raising it creates unemployment; certain jobs become too expensive (the cost to the employer of paying the wage) to be worth paying anyone to do it. Furthermore, not all people who "work" need to "make a living to support a family." There are always young people who do not have a family to support, they are still relying on support from their own parents, and they want to get a beginner's job in order to gain work experience that will help them find better, higher-paying jobs when they do move out on their own.
"Republicans are all for 'trickle down economics.' "
Oh? Is that what you have heard Republicans claim they are for? Or is that what you have heard Democrats accuse Republicans of being in favor of? Do you understand that the expression "trickle down economics" is just spin created by Democrats to misstate the Republicans' position?
"They like to claim that increasing the amount of money the wealthiest people have will trickle down to everyone."
No. Republicans believe cutting taxes on wealthy people will open up more opportunity for wealthy people to start new businesses and expand existing businesses and doing so means more jobs for more self-identified "working people."
"So why don't they support an increased minimum wage?"
I told you above.
"After all, increasing the amount of money the poorest people have will trickle UP. Poorer people, when earning more money, will almost definitely spend it, increasing the economy."
Gee, why not just create a minimum wage law that every "working person" must be paid at least $100 an hour? That will surely make absolutely everyone super-rich, won't it?
"Many wealthy people take their massive tax cuts and hold onto the money"
Oh? Have you counted how many will do that (because that's all they've ever done when taxes have been cut)? Do rich Democrats like Michael Moore do that too?
"So why doesn't the GOP support increased minimum wage, something that would work much better than the oft-tried, repeatedly-failed 'trickle down' joke?"
The "joke" is coming FROM Democrats who do not describe what Republicans are doing accurately or fairly. And I've told you why Republicans don't want to increase the minimum wage - it's foolish economics; not everyone single "working person" has the same minimum economic needs, and raising the minimum wage eliminates jobs. It also leads to greater inflation. Prices have to go up if employers have to pay their employees more than what the employers calcuate is the value of the labor.