They've been accessible for a lot longer than most people think. The original Minimig project (an FPGA recreation of the Amiga chipset, coupled with a real 68000 CPU) started in 2005 - more than 20 years ago! And 15 years ago there was already a complete Amiga core (chipset and CPU) running on the Terasic DE1 development board, the C-One FPGA computer, and the Turbo Chameleon 64 cartridge.
Today's FPGAs are certainly more affordable and more capacious (especially in terms of DSP and RAM blocks) but the biggest shift is that, as you say, it's now possible and affordable to have the complete PCB assembled in short runs, which is a real blessing given that so many FPGAs come in BGA packages.