Your arguments are interesting, but they hold only in systems with artificial inflation controls. UBI increases the amount of money in the economy, which has the same effect as simply printing more, thus the value (scarcity) of the money decreases. A poor mother without UBI will be a poor mother with UBI because costs will simply go up, something you acknowledge.
As costs for services go up, costs of products go up as well. For example, if I'm in the beer business, and the cost for me to bottle a single beer is $.10 per bottle on my assembly line. If labor costs increase under UBI to double that ($.20/per bottle) (because cheap labor is now harder to find), I must increase my sale price by at least the same or go out of business.
Now somebody buying my beer has to pay $.60 more per six-pack, potentially wiping out that fraction of new income UBI was providing them.
The government could come in and say "the price of beer may not increase at all" and set some kind of price control. So now I need to cut $.60 of cost somewhere else in my product.
But wait, it gets worse!
Bottling isn't the only cost for me to make and sell my beer.
- Grain harvest is more expensive - increasing my cost
- Transport costs are more expensive - increasing my cost
- Blank bottles are more expensive - increasing my cost
- Brewing is more expensive - increasing my cost
and so on...
So to prevent inflation I have to cut costs everywhere else. Cheaper glass, worse grain quality, worse brewing methods, worse water supply, and so on. If I can't balance the cost equation I simply go out of business, decreasing the supply of products in the economy and increasing unemployment.
In effect, nobody gets paid more because everybody gets paid more, which drives up prices, which is the definition of inflation. If inflation is artificially capped by price controls, then product quality either goes down, or I go out of business.
It's basic economics.