Why do cycle paths always have to be bumpy?
I think it is an issue of foundation mostly. Because roads get very heavy (as in weight) traffic roadbuilders spend a lot of time on the groundwork, compacting, digging away soft and unstable earth or sand layers and replacing them with gravel or other heavy rocky debris before the tarmac is laid down.

Because a bicycle path only sees light bicycles less time has to be spent stabilizing the base before the tarmac can be placed. Unfortunately this means there is less of a hard pack underneath the pavement to keep roots away, and there is not as much heavy rocks and other stuff underneath the pavement stabilizing inevitable small shifts in the ground. So bicycle paths shift a bit more, and they are more vulnerable to getting pushed up by roots.

What it comes down to -I think- is money. Building a solid foundation takes time and materials, and hence money. For a bicycle, anything paved is better than what was there before (nothing or running the gauntlet on the open road) so I assume municipalities choose to get the bicycle paths out there fast and cheap rather than roll them out slow but more expensive. Spending a lot of money on car-load resistant bicycle paths also just doesn't make economic sense for most municipalities.

The good news: Here in the Netherlands a lot of municipalities have realized over the years that the cheap bicycle paths cost a lot more in maintenance and repairs, so the quality is steadily improving.