Even if the FPGA design were fully public, you won't necessarily find the required fab around the corner, so what's there to gain from the FPGA design? If you're seeking trust, you'd need to establish trust in the whole chain: design, mask production, chip fab, transport... that's a tall order.
The argument here usually is: with a fixed silicon chip, vendor can hide the backdoor in various locations and be it triggered by various events (e.g. a particular sequence of incoming ICMP packets would overwrite the first byte of response with content of some register). With FPGA, the vendor can't really know where a particular register is located, or where incoming packets are processed, as it is highly dependent on the synthesised CPU design and can even be non-deterministic.
This does not mean that there is no way vendor can backdoor the chip you are getting, but it does narrow the possibilities significantly.
The usual thing the military worries about is a "kill switch" (a very unlikely sequence of bits) which disables the hardware completely. The idea is that at the beginning of a war, the kill signal is broadcast by the enemy by every means possible which brings all your electronics to a halt.
This can be hidden in an FPGA - for example attached to the input pins or SERDES - without needing to know anything about the application.
Triggering a malfunction is incredibly easy compared to a proper backdoor. A kill signal could also be injected through side channels e.g. a power line, and the kill mechanism could be implemented in many other semiconductors than an FPGA.