You could certainly use a microcontroller to implement arbitrary digital (or even analog) logic within a circuit. You'll be limited by the CPU speed. The trick is to avoid interrupts, and spin the CPU in a tight poll loop instead. An Arduino Uno's MCU can run an instruction perhaps every 150ns on average. You could implement some simple logic that runs upwards of 1Mhz. If you use a cortex-m CPU, you could do fairly complex logic at several Mhz.

That's a silly way to use an MCU, though. You're supposed to use the peripherals to offload enough complexity from the CPU that you have plenty of cycles to implement your application logic in a straightforward, maintainable way.

Actually you can get the same experience with an FPGA if you pay for the FPGA development tools, maybe what you’re asking for is a low cost version of Quartus or similar tools. I would like it if that existed but I fear FPGA makers deliberately want to keep their tooling expensive and hard-to-learn to promote lock-in.