A few nand2tetris fanatics have actually done this! And by a few, I mean quite a lot of people. Here's one such hardware project of nand2tetris: https://gitlab.com/x653/nand2tetris-fpga/
I kind of want something midway between the FPGA version and the all-transistor version, something that just uses 7400 series chips (or, presumably there’s a 26-pin equivalent with 6 gates instead of three). Heck, I think even something that goes ahead and uses the full panoply of basic logic chips available could be kind of cool to see.
It's been a few years since I studied it (I even built the clock module, registers and the ALU), but from what I remember the biggest departing point from what you want is that the control logic (from instruction decoding to deciding which sub-units to activate for each instruction) is done with an EEPROM instead of individual logic gates, as described here: https://eater.net/8bit/control