What you need to do is drain all power from the machine, by unplugging it and either waiting for a while (at least 5 mins) or pressing the power button a couple of times (this attempts to start it, which will drain any capacitors in the PSU – typically, fans might spin up or LEDs might light for a bit).
This is an interesting quirk of how different motherboards (specifically, the BIOS/UEFI firmware) handle displaying the menu when hibernating. ACPI defines a special power state for hibernation, S4 (under G1, as are all other standby modes), which the BIOS can decide to handle differently. for example by booting from the same device without displaying a menu.
What you need to do is put the machine into the powered off state, S5 (under G2, normally entered when ‘shut down’). The easiest way is by going to G3 first, by removing all power.
There does not appear to be an option within Windows to tell it to use S5 for hibernation – but then, that’s what S4 was designed for anyway. Windows is not doing anything wrong. It may be useful to take a look at your BIOS/UEFI options/manual, since there may be an option to change which state the computer enters when an S4 signal is received.
The only real way to ensure that your next motherboard has this option or always displays the boot menu is through experience, whether yours or others’. This is not usually an advertised feature.
One thing to note is that the BIOS (I’m not familiar with the UEFI booting process) might load a certain device (e.g. a hard drive), but it is up to the bootloader on that device to resume from hibernation or provide additional options. The Windows bootloader will always resume when hibernated, as far as I know. I believe GRUB may be configurable to provide the boot menu every time.