The power profile of an FPGA is very different (worse) than dedicated silicon. Both the background current and cost of a gate switch. There are a lot of situations where that isn't acceptable.
True, that's definitely possible. I was giving the habitual answer (that I occasionally need to explain to a client at work) why you need to spend $2m having an ASIC made when "it's just digital logic and we got the prototype going in a month"
They usually care about at least one of - size, power, cost.
So realistically, if you need fully custom digital logic, you'd buy LFE5U-12 instead and program that.
So that's $16 FPGA from widely available distributors (like Digikey) who likely can afford 1 or 2 day shipping.
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Custom chip design for a flex-circuit is interesting, but only if you have substantial analog parts that cannot be easily implemented by an FPGA.