monitor(s) intermittently failing to activate on cold boot up

I’ve started encountering a problem with my desktop PC running windows 10.
I have 2 monitors fed by a GeForce GTX 670 on a Gigabyte GA-Z77X-D3H motherboard with an Intel I 7 3770K chip.
Recently the monitors have failed to display anything when the PC is switched on – not always – but frequently enough recently to raise the alarm that this is a problem.
The PC emits its usual beep but no start up screen or any other sign of boot up occurs with increasing frequency.
I’ve tried booting to ‘safe’ – no effect ; un/replugging the hdmi cables and swapping them over – no effect ; soft rebooting and completely disconnecting from the mains. A total switch off from the manins worked twice but now even that cannot guarantee a cure - the failure rate of booting to a visible desktop is getting greater.
This is a fairly venerable machine (built 2012), but has performed as a steady workhorse for years, requiring only a thorough vacuuming some time ago to deal with filth build up on internal fans.
I might add that when (as now) both displays are on, everything is normal, no shudders, odd noises, nothing suspicious at all.
Before I dismantle things, has anyone a pointer as to the most likely cause of this gremlin?
BTW – 1 further point.
The button battery on the motherboard. If this is replaced (and I suspect it’s on its last legs), does it mean that anything has to be reset? Over the years I’ve fitted 3 extra hard drives.
I hate going into BIOS, the Gigabyte GA-Z77X-D3H motherboard display is not a model of clarity and a lot has been lost in translation. If I can get away with not having to navigate the BIOS screen, either in its traditional or Gigabyte’s flaky schematic form, I’d add years to my life

PSU possibly has failed caps and oddly enough, caps of high-quality manufacturers seem more likely to be invisibly failed. (no leaking or bulging to be found)

But caps of good quality, usually are still OK.