When I was grossly underemployed I split my time between finding a new job (which I agree should occupy a large percentage of one's time) and working about 30 hrs a week to help keep my sanity and pay bills. My choice. I had out-of-the-area offers that were slightly better than what I have now, but with the accompanying costs of relocation. It was my choice to take this current position about 6 years ago at a lower salary and not have to uproot my (then) teenager and older one still in local college living at home. Since then nothing better has opened up in this area (in fact many more jobs have been lost) and I have still not looked outside. Late career complacency I guess. Again, my choice.
How to divide one's time in any case is still a gamble. You can forego all income but the UC and focus on the search, hoping you find something before the benefits run out. Or you can cushion that some and do both (search and work). Individual situation dependent.
In THIS state an employee pays into the UC fund, albeit a lot less than the employer's share. Might be true in other states as well. Regardless, any time an employer pays a tax on behalf of an employee, it's that much less that he could (not saying should or would) be paying that employee. Not participating in the available bounty was my decision for reasons I stated, but that also delayed available funds. I don't chastise anyone who decides differently. But I do have some issue with those who milk the system...