Many local producers can continue producing their ice-cream or whatever it is they produce.
Not quite true. If they don't have the right paperwork, they won't be able to sell legally. I know that from next year, charcuterie-makers will have to comply with EU regulations (they've been exempt for the last couple of years to give them time to modernise). Many don't make enough to warrant the investment required, so they will either fold or go underground. The same goes for cheesemakers.
The EU's prime objective isn't to force such companies underground.
Of course it isn't! But that's the effect.
they need to maintain certain standards and strive for harmonisation.
Sure. But it needs to be appropriate. You can't apply the same standards for a small family-run enterprise that you do to a national manufacturer; many family businesses have been producing food for a long time without poisoning anyone.