There are countries that 'action' this exact policy. Take Australia for example. Unemployment payments are based on the concept of mutual obligation, where those receiving unemployment benefits must demonstrate that they are searching or training for future employment: https://www.servicesaustralia.gov.au/individuals/topics/mutu...
Those who are unable to work, based on certain medical eligibility criteria, do not have to meet mutual obligation requirements to receive the disability support pension: https://www.servicesaustralia.gov.au/individuals/services/ce...
> You can also contribute to society in ways that are not defined as work. What happens to someone that is taking care of children, or taking care of their sick parents? Should they be assisted financially, or not? If so, how do you define that?
Again, this is a policy question that is answered in Australia and which is implemented administratively via the 'carer payment': https://www.servicesaustralia.gov.au/individuals/services/ce...
I agree with your point that people will fall through the cracks and that there will be mistakes in administering any system like this. But that is simply the nature of a rules-based bureaucratic system, which trade off the ability to handle edge cases with scalability.
1. It's, in part, a thinly veiled program to transfer public money into the pockets of privately owned "Employment Service Providers"
2. It's offloading the "checking" to other private companies by way of requiring people to apply for a certain number of jobs. Any jobs. Maybe the same jobs you've been rejected for again and again so many times that HR recognizes your identical cover letters coming in every month.
3. Might be an extension of 1, but Employment Service Providers (private, for-profit companies) have the ability to report you to the government and get your payment cancelled for missing certain appointments, with little oversight and no repercussions for errors.
The whole employment services provider model is a weird frankenstein pseudo-market which operates according to bureaucratic rules and artificial 'prices'. And this model requires the bureaucracy to monitor the pseudo-bureaucracy (employment service providers) to ensure they're delivering employment services appropriately. The thinking behind this model is similar to arguments made in favour of privatisation - that private firms will always be more efficient than direct government service provision.
Lots of policy trade-offs to consider: Should government directly provide employment services or is a pseudo-private market more efficient? Should we just drop mutual obligation entirely at the cost of potentially higher structural and hardcore unemployment (and so higher government outlays on transfer payments)?
The devil is in the details. I could take this viewpoint and take several trade-offs on the implementation which would result in either extremely left-wing or extremely right-wing policies.
However, that is not what politics is (supposed to be) about. It is supposed to be about actionable policies and their effects. When putting this into an actionable public policy you run into many problems. How are you going to police the "except if they cannot work themselves" part? What are the requirements for "not being able to work"? How lenient are you there? What happens to people that fall through the cracks because of mistakes made in policing that (i.e. people that cannot get a job, but do not qualify for the help)?
How much money does it cost to police that, and what are the costs of not policing it at all? If the cost of policing is higher than the cost of not policing and just accepting that some people will take advantage of the system, is it still a good idea to police?
You can also contribute to society in ways that are not defined as work. What happens to someone that is taking care of children, or taking care of their sick parents? Should they be assisted financially, or not? If so, how do you define that?
Then there is the issue of automation. Perhaps people are not entitled to the fruits of other people's labor. But what about the labor of machines? As more and more jobs get automated, who is entitled to that? Should the owners of the machines reap all of the benefits while the rest of society is left with less and less as more and more jobs get automated away, or should this be divided amongst the total population?
Different people will answer all of these trade-offs in different ways, even if they may agree on the core view. Left-wing and right-wing people generally share most of their core views. Nobody wants more crime, more poverty, etc. The implementation of these policies are where disagreement and discussion happens.