Lots of reasons. I'll list the ones I can think of off the top of my head. Some of these reasons are to do with your own personal circumstances, i.e. the LA may treat you worse if you submit samples than if you don't. Other reasons are more community/politically motivated, and are to do with establishing expectations that an LA won't overstep its remit, in just the same way someone might refuse to comply with an unjustified police stop and search which appears to be to do with racial profiling, though it might be easier to comply on this occasion because you "have nothing to hide".
Legally, it's a question of provision rather than the child's performance.
For an individual child, there can be no benchmark against which to assess the child's attainment other than the child's own capability. The LA isn't in a good position to know the child's capability and whether they are meeting their potential. Measuring attainment makes a little more sense in the case of a school, where there are multiple children and one can consider whether the group is underperforming statistically compared with other groups of children.
Some families don't do school-style work. If those families who DO do school-style work submit samples, it may create an expectation in the LA that all families should educate in that way. LAs are notorious for saying things like, "other home educating families are happy to provide xyz" with the implication that doing otherwise is dodgy.
If a parent is conscious that work samples will be submitted to the LA, they may put inappropriate performance pressure on their kids to produce something good. Parents get anxious about this. It feels like there's a lot at stake, especially in cases where school was disastrous for the child. Think of the pressure which many schools put on Y6 kids to perform well in SATs, and then imagine how that would be intensified if the school were to be judged by the performance of just one single child. Kids who were traumatised at school can live in fear that they will be sent back if their work isn't up to scratch.
Some LAs who receive work samples will subsequently deem the family to be failing if insufficient progress is shown next time. But children often have periods of no apparent progress followed by periods of rapid progress. One of my kids was exceptionally good at maths aged four, then set it aside for years before having another burst of interest and dramatic improvement from 9-12 (but focused on concepts rather than arithmetic), did little with it until GCSE and then blitzed through the whole school curriculum in a few months. I don't know what the LA would have made of that! Such patterns are pretty common with individualised learning.
Some people regard a demand for work samples as indicative of distrust on the LA's part. If I report that my child can confidently add numbers up to 100, asking to see a worksheet on which she has done so feels like an inspection rather than the informal inquiries which LAs are meant to be doing at this stage. And it would be easy to fake, so what's the point?