Then why do the suppliers are intent on lobbying that they have to provide the least amount of information that's possible?

1. Supplying useless information still has a cost just because supplying any sort of information is not free.
2. Some consumers are irrational and supplying some information plays into the biases of the consumers
3. Some information the government may want manufacturers to supply for political reasons unrelated to the safety of the product. Imagine a law which required that manufacturers state whether the product is made using labor from illegal Mexican immigrants. This can overlap with #3.
4. The manufacturer may fear that consumers will interpret required labelling in the context of non-required labelling--for instance, if consumers are used to safety warnings, they may be more likely to interpret some kinds of required labels as safety warnings.

It's a bad thing, but still not an externality.