Edited to extend: this is why TCP has a "Maximum Segment Size", and why Path MTU Discovery information has to be passed into the TCP state machine. It is TCP that takes responsibility for carving up the data into the packets, not IP.

One of the goals of UDP was to avoid needing this kind of state, which is why the IP layer handles fragmentation for it instead. This is allowed on a hop-by-hop basis, unless the DF bit is set; so when a "too big" packet gets to a node with a smaller MTU, it can just split it and send on the fragments. No PMTUD needed.

The design could have been for the fragmenting node to also add a UDP header as part of that process, but was not. It would have been a simple change at the time. It's had a lot of consequences since and is responsible for a decent amount of complexity in hardware and software packet pipelines.