FPGAs are often used to test/design not a "circuit" but an ASIC (application-specific-integrated-circuit) that you will then go on to actually build. Or, if your application doesn't have volume to suppprt the ASIC engineering costs but can support the FPGA unit cost, you just leave it as an FPGA. There are millions of devices (industrial machines, research, testing, etc) where this makes sense.
And not all FPGAs loose state when powered down. For those that do, there's typically an easy way to hook up a little Flash chip and bootloader to reload the state automatically on reset. For those that don't have that option (EG, currently working with an ABB robot, Mecco laser, and Rockwell PLC which all contain FPGAs, only the Mecco can boot itself), you store state in battery-backed SRAM and just leave the RAM on at all times.
In addition, you get the opportunity to both update the part and/or to change it's behavior via (for instance) a user setting.
To be pedantic, an ASIC (and any IC really) is a circuit, as denoted by the "C" standing for "circuit".
We use FPGAs in cell sorting because we need to make decisions off of high dimensional data with low latency. The cells moving through our system have velocities higher than 1 m/s. They flow past a set of lasers and wind up in a droplet less than a millisecond later so we need to make a decision whether or not to sort a droplet within that time frame. We don't use ASICs because we don't move enough volume to justify the startup cost.
I bought all these things off the shelf. There are definitely some markets that are well served by FPGAs but are still too small for actual fabrication processes.
Which isn't to say every programmer shouldn't be forced to struggle through a Verilog or VHDL project - that'll put some hair on your chest and the fear of god into you. I'll see your async and raise you an "always @(posedge clk)"
I personally know of many embedded systems using FPGAs (spacecraft and medical applications), signal processing such as radar, many PCIe cards, etc.
https://www.pcper.com/reviews/Graphics-Cards/ASUS-ROG-Swift-...