I thought custom ASIC's were done when you have some computation that takes many instructions to implement in a CPU, for example for Bitcoin mining your computation might be "Find the sha256 of a fixed-length input and tell me if it begins with at least 32 zeros." The main benefit of using an ASIC (as opposed to an FPGA) is you can hard-wire the circuits, the FPGA has to use generic hardware (super-flexible LUT's and a super-flexible routing fabric) and you pay for all that flexibility with speed, power consumption and area.

I guess the main benefit of the chiplet idea is that you can get a really fast, low-latency connection between the CPU and FPGA? What kinds of problems can be solved with that architecture, that can't be solved by just putting the CPU and FPGA as separate chips on the same circuit board?

Or is it cost and design simplicity, you can have a lot of chiplet elements on the same chip, and the customization is in which chiplets you want ("give me a chip with 4xFPGA and 6xCPU") and how they're connected ("CPU A should be connected to FPGA #4, CPU B should be connected to FPGA #2 and #3")?