Actually, if a state does any welfare, it probably has this problem already. Basic income offers significant reductions in those inefficiencies by simplifying the process of selecting eligible people (and retiring the required bureaucracy).
> and make sure politicians get their hefty share when the money flows
That you have under every system, from despotism to socialism to capitalist democracy.
> Inequality is a false problem. It's a problem if people get poorer, but all indicators for the past 50 years show that everyone is getting richer, while rich ones are getting richer faster (which is obvious why, when you have more capital you are much more likely to increase it faster). As long as even the modest classes are getting richer every generation, everyone's life is improving.
Inequality will be a false problem when the bottom level of society won't be starving to death or due to lack of basic health care. Again, basic income proposes turning inequality into a non-problem by securing a decent, humane minimum level for everyone.