What is the modern way to do small scale programmable logic?
The modern way to do such a digital logic circuit would be in an FPGA or CPLD.

The way I'd recommend for your particular logic circuit is a CPLD. Have a look at the Altera MAX10 family or the Lattic iCE40 devices. You can buy a cheap demo' board like the iCEStick, download the free development tools and get something up to experiment with at home or work.

You will have to learn VHDL (my preference) or Verilog but that's part of the learning you mentioned you're interested in doing.

You could use a microcontroller to produce a similar result but by a quite different function...but it doesn't address your question. And you'd have to learn to programme that so there's work either way.

(I know Lattice call iCE40 an FPGA family, but across the logic chips market, they have more in common with CPLDs. The name will do for the purpose here.)