That right there sounds prone to abuse and unfairness if applied to any significant extent outside of a limited experiment. Expect shitloads of "unmarried" or suddenly "divorced" couples claiming $13k each. Kids need to eat and go to school and can't earn for themselves.
The only way this could possibly work at least to some extent as a replacement for welfare without its accompanying bureaucracy is if it's straight $13k for everyone, not dependent on marital status or age limitations, but you would still need some government involvement to make sure that e.g. kids are not starving and people with disabilities or other issues have a fair chance. Unless we're disregarding all that as part of the whole welfare system, in which case we might as well cut the UBI off at the age of 65 and let them old-timers fend for themselves.
I like your straightforward approach but I do not believe that children should be considered in the universal equation as "equals". I would not be opposed in any way to having a smaller portion allotted for a household with children as there is a substantial cost increase associated. Believe me I know

I also believe there will still need to be considerations taken for the disabled and elderly, to ensure proper care. Especially when both groups can at times be mistreated, forgotten or taken advantage of.
Again the UBI, is not meant to be something you can make a living off, it is supposed to help everyone have the opportunity to meet the basic necessities of life, but not being limited to lower income earners. It also is not tied to only being able to spend the money in a certain way, it is your income, which also has the potential for abuse. I just rarely any program should be tweaked based on the small number of people abusing it. The fine tuning of how benefits are calculated and paid, should be something we can learn from these experiments if they are run start to finish and analysed in an unbiased manner.
To touch back on the shitloads you describe, yes there may be something along those lines. I imagine then that they would have to be looked at tin the same way as a common law relationship is. Which most of the time is no different than married. The added cost of a new address would negate the gains these people may be trying to grab.
It is also currently happening or at least being considered.
https://www.today.com/health/happily-married-couple-considers-divorce-pay-daughter-s-health-care-t132915