It did help with reboots by pressing the reset button. Now card mounts ok with every single reset 
Still, powering SEA with SD card inserted doesn’t work with one specific card I have.
If the board is powered on with the card inserted, I get this error printed repeatedly:
If I plug the sd card after the board is booted, and press reset - then it boots fine.
Not only it doesn’t boot on power on that way, it also fails the programming from within Arduino IDE while the card is inserted.
Apparently, external/forced pullups are interfering with the ESP32 bootup sequence. Pulling the MTDI (GPIO12) pin high at boot (/power?) time selects the 1.8v spi flash interface. And while SAE isn’t using VDD_SDIO pin to actually power the flash chip (according to schematic the flash is powered with the main 3.3v rails), the selected 1.8v voltage should be still used for SPI interface buffers, and shouldn’t work properly with 3.3v chip.
In addition to sdmmc_host_pullup_en() call update, I’ve burned fuses with espefuse.py, forcing the ESP to select 3.3v for flash, ignoring the GPIO12 pin. Looks ok now It starts somewhat longer on cold power on, but it boots, and the fpga is configured without any manipulations with sd card or reset button.
Programming with Arduino IDE is also ok while that “troublesome” SD card plugged in