Here's the main problem it would solve.
Fathers could earn enough on a regular full-time job that mothers could stay home with the kids.
Single mothers could earn enough on one job that they don't have to work multiple jobs or long hours.
My dad was raising us on $9-11/hour back in the late 1980s and early 1990s, when the minimum wage was $4.25, so I'm guessing that's about equivalent to $15/hour today. With that he was paying off a cheap mortgage in a low-cost area and we were never below the poverty line. Sure, we didn't have extra shyt and money was always tight, but we made it. School wasn't shyt, but I graduated with great grades and got into college. When us kids got old enough to take care of ourselves, my mom got a job herself and started working part-time, then eventually full time. My dad eventually got a promotion and was able to retire just fine. My mom will retire soon, having worked full-time for 20 years, and they're doing quite well. All three kids went to college - my parents couldn't really help much with that, but we got scholarships and financial aid packages are fine in that area.
My dad was able to make enough for the whole family and my mom could stay home. That's a BIG difference in life. If every head-of-household had a stable income and every young kid had a mom at home, 1/2 of the community's problems are getting solved there. Wages not being high enough to support a family is one of the reasons that ain't happening.
A living wage doesn't solve every fukking problem in the world. But it doesn't have to. We're not saying that everyone will be stuck on a living wage or on one income their entire fukking lives. But there needs to be a baseline that people can get through that early family-rearing stage on their own, then they have another couple decades to worry about retirement with.
You realize that a quarter of income for someone making $48K and a quarter of income for someone making $15K are different things?
Take a quarter off of $48K, you still have $36K left to take care of your other needs. Take a quarter off of $15K and you only have $11K left. That's a big difference.
Can you find one example of someone buying a new house right now on minimum wage income? Like, one example in the entire fukking country?