OP, I can empathise with your feelings, having had two relatives claiming benefits in dubious circumstances:
One received disability benefits for decades, the relevant condition relating to a bad back. We never saw any evidence of back issues that meant that they were unable to work. Later in life, they did some training (I recall, rather ironically, in a physio-related area) and ended up working in the health system for a period.
The other received unemployment benefits whilst also working cash-in-hand in tradesman jobs, amongst other pursuits. At one point, this person was doing some work on our house and said they wouldn't be able to attend in the morning because they had to go to the dole office to collect their cheque! I would be surprised if this person ever submitted an accurate tax return. The closest I came to contemplating dobbing them in was when their house, which was on a very large block, was largely destroyed by fire. I do not know whether they was receiving benefits at this stage, but they still seemed to be enjoying lots of 'off the books' income. In claiming insurance for the fire, they included lost income from tenants who also lived at this property. I wondered whether the insurance company would be interested to know that these individuals were not in the fire-affected house but in separate cabins, and whether the Tax Office would be interested to hear about all this rental income!
OP, I never did anything, and indeed never seriously contemplated reporting either of them. But as someone who is a firm believer in having a safety net, and who has (happily) paid a lot into it, it was frustrating to see relatives deciding to opt out of the social contract. I couldn't understand the ethics of it, and it seemed so selfish. I guess they just had a very different view of the world.
Should you feel like a 'bad person' for not acting? I think some of the problems have been highlighted by others, including whether you have access to all the relevant evidence. But even if your relative is gaming the system (as I'm sure one of mine was, and likely the other too), I also accept that (1) there will be a certain amount of fraud in any benefits system and (2) it will be too costly and counterproductive to stamp all of it out. To the extent there are, in my view, huge issues with the setup and sustainability of the benefits system in the UK, I think that there are other, far bigger, issues we need to address than dodgy claims.
I hope this helps, OP.