Bits field vs actual difficulty?

[sashavich04]

In BTC, each block has a Bits field with a rounded representation of the actual difficulty, which was calculated from the last 2-week adjustment.

Why do blocks have this Bits field in it?
Everything else about blocks feels like they are calculated whenever possible, keeping things compact with no redundant data.

Doesn't having a less precise version of the true difficulty cause problems? What happens when a new block is generated, which is valid according to Bits, but is not good enough to meet the true difficulty?

I am confuse.