It's a wash. If you "give the money to everyone equally", you'll just be clawing it right back by raising taxes on higher-income folks. Theory suggests that you'll want to phase out the aid quite aggressively as income increases (though not by anything near 100%) because this is where the tightest constraints on spending are no longer binding, so the potential benefit is not that worthwhile.
One is having to prove you are under threshold (documentation, paperwork etc) - the other, everyone gets it, so much less overhead for the poor (the overhead is on the side of the people who can afford it - the VAT tax or whatever is on the other side)
This is a tax credit, so there ought to be very little paperwork involved other than filing for your taxes - I agree that this kind of arrangement makes sense. It's just that describing it as a subsidy for lower incomes is inherently fairer and more accurate than trying to make it into something "universal".
I know the answer, no need to reply to me. Regardless. Give everyone equally.