No, flashing the card won't help.
The work around is to run Luxmark for 10-15s after boot. You can auto load that after login if you want to. It's a known bug for 4,1 or flashed 5,1. It seems SMC related (that's why real 5,1 is not affected).
Another work around is use MacsFanControl to setup your own fan profile. e.g.
PCIe fan base on PCIe ambient, min 38, max 58
PS fan base on PS 2 temperature, min 36, max 58
Since MacsFanControl alter the fan profile by inserting the target fan speed. That will completely override the SMC fan setting.
Of course, you can use / write any software to do that as well as long as you can control the PCIe and PS target fan speed. However, AFAIK, there is no fan control software alter the target fan speed like MacsFanControl does. All others are altering the min fan speed, which is good for general use, but unable to fix this high idle after boot. Therefore, you practically has no choice unless you write / mod another fan control software for your own use.
If you only boot once every few days or even weeks, I suggest the Luxmark method. (Unless you want a cooler PSU etc)
If you boot few times a day, then MacsFanControl method should work better (but it will take up tiny bit CPU resources).
You may also try SMC reset. I tested this method. If you keep trying, the fan will eventually go back to normal (may need >10 attempts), but the bug usually come back in jut few reboot. So, not worth to do so.