Obviously it's a complicated issue and the minimum wage is just one option among many which requires co-ordination of multiple policies just like all the other options that have been mentioned whether they are new or not.
The minimum wage has been disproportionately blamed for issues that are due to a lot of other factors, some of which may be minimally affected by it, singled out as if it exists in a vacuum or as if it should work when mixed with other policies that contradict it, and most definitely politically demonized, if that same kind of evaluation is put upon any other policy they would appear ineffective as well.
"Trickle down" economics is based on the premise of beggars and lack of freedom at the bottom thereby they must bear with the trickle, lack of a minimum wage facilitates such a setup.
Simply if the goal is to get people to look for work then the work has to pay enough.
Again, your suggested remedy (high minimum wage) doesn't really address any of the ills you stressed as need fixing.
Minimum wage doesn't deter work, entitlement does. There is no minimum wage in China or most (any?) developing economies, yet you don't see millions of Chinese people kicking back saying it's not worth working do you?
Minimum wage is really only a thing because the western welfare state made it so. That if the basic state wealthfare one gets for doing nothing is equal or higher to the wage one earns working, that creates a perverse disincentive to go out and work, so a minimum wage is needed to ensure even the lowliest work pays more than state wealthfare.
The problem with that is that it's a vicious circle.
You increase the minimum wage, which fuels inflation, which means you (as the government) are under pressure to increase wealthfare payments as the purchasing power of existing welfare allowances are eroded by said rising inflation. Raising welfare payments again creates a disincentive to work, forcing another minimum wage increase. So on and so forth.
I do agree with you that there are a lot of prejudice and bad policies that unfairly disadvantage poorer people, who often get dealt such a poor hand by simply being born poor that no amount of hard work could get them out of that poverty trap.
In my view, a key component of the disincentive to work is also a realisation and sense of resignation on the part of the poor that 'this is my lot in life'. That the 'game' is so rigged against them that no matter how hard they work, they will never break out of the poverty they were born into, so why bother at all? Better to just enjoy life as much as possible.
That is the area where the government should be focusing its resources and energies - in eradicating barriers to social mobility and giving all citizens a fair opportunity to make something of themselves through hard work and dedication.
If you contrast the poor in places like the UK and China, the most striking difference, other than differences in state social welfare, is the general sense of oppitism (or lack of it).
In China, there is the real belief and expectation that one could improve one's station through hard work. If not for oneself, then at least for the next generation.
It is that promise that made migrant workers leave their homes in the tens or even hundreds of million to work in the cities, where there is no magical minimum wage to motivate them.
Again, in the China example, western pundits and analysists love to stress China's 'poor' workers' rights situation, and yes, there are indeed too many cases of exploitation and poor treatment. But those are still the tiny minority that gets disproportionate western media attention.
The unreported fact is that for the overwhelming majority, even with minimal government regulations, both working conditions and wages rose significantly over the years, so much so that China is now fast loosing its once indomitable labour cost advantage compared to western developed economies.
That wasn't achieved through minimum wage, but good old supply and demand market forces.
As I said, price setting is one of the things free market does reasonably well.
The core problem with trickle down was that it was based on loony toon wishful thinking (at best, more like lies bought and paid for by the rich in reality) economic theory that bares little to no connection with real world events.
The true effect (arguably aim and goal to start with) is to rob the poor to feed the rich and to protect the position and status of the rich against encroachment by the upstart peasants.
The condescending hostile attitude the proponents of trickle down towards the poor for having the gull to be born poor and not know their place is a core reason for the disenfranchisement and lack of motivation on the part of the poor.
However, even a broken clock is right twice a day. Just because trickle downers advocate it, does not make the argument against a high minimum wage automatically defunct.
One needs to carefully examine each policy by its own merits and disintengle it from all the other political ideological nonsense the press usually packages it with.
As such, my conclusion is that a high minimum wage is a highly ineffective, and ultimately self defeating way to address the core fundamental wrong we both agree on - lack of social mobility and inequality in societies.
My argument is base purely on the practicality of the policy, not the loaded ideology opponents of minimum wage often air in the western press (owned by the rich I might add).
Simply put, there are far better ways to go about boosting social mobility and reducing inequality. And as a demonstration of just how crap minimum wages are, literally handing out free money would be a better policy in my view (not the best by a long shot mind you).