I think your diagnosis is spot on. Capacitor most likely, since they store charge and can fail. There is a happy ending to this sad story. I spent 2 days looking for the receipt, without luck. Finally I called the store and they produced a copy from their system, from 2014. Warranty expires in July! Couldn't believe it. They have asked me to come in and they will warranty the board. I don't know what ASROCK will do, since the board is out of production. I guess they will have one somewhere, or fix it. 

The down side is that I'm in the middle of recording an album and this is a big disruption. Installing all the software onto a temporary computer doesn't appeal to me - it takes literally days. I don't want to buy a new computer right now, with the i9 probably around the corner - plus this one is still running fine.

Thanks again. I think it's a shame about the lack of direct support. Under similar circumstances for another project, I was able to contact Gigabyte and they replied immediately and fixed a board out of warranty, for free. ASROCK should be more interested and not relying on forums to help people.

That's not a solution. That's a hack. I would like some attention and an RMA:

https://youtu.be/CExDnNXuJQs" rel="nofollow - https://youtu.be/CExDnNXuJQs

These forums exist so users (which we moderators are too) can try and help each other with issues, often times this is a quicker and simpler way to work through a problem than waiting on Tech Support. A lot of issues are common place and arise because of incorrect assembly, setup or component choices. When this is the case the forums are an invaluable tool. 

Sadly, not all issues fall into this category, sometimes you need the specific product knowledge, tools and skills of official Tech Support or to contact your place of purchase for RMA instructions. 

As for your issue, systems today are designed to remain powered 24/7. Rather than cutting power completely the system will retain a small amount of activity that would barely measure on a power meter. This saves components from being damaged over time by repeated influx current when powering on at the wall/power brick. Using the system in this way is not a hack, it is using it as it was intended to be used. That said, the system should also be able to handle a true cold boot from power loss (power switched off to the system) without causing the issue you are experiencing. Something is not working correctly and you are well within your rights to want it resolved. 

You do have an issue, this is not normal behavior, if it were, we would see numerous people complaining that their systems fail to post when they first turn them on after a build or even a simple loss of power. I am sorry if you felt your issue was not getting the attention you hoped for but you must understand we can only help with issues we have experience with. Strange behavior like this is uncommon. 

For what it's worth, here is what I would do to troubleshoot your issue.

Disconnect everything except for POST essentials, 1 module of RAM, CPU and PSU and connect only a monitor, no USB devices at all, not even mouse and keyboard. 

If the system still takes ~20mins to POST from power loss then start checking connections from your PSU to your board, given you have already replaced the PSU I would examine the 24pin and 8 pin connectors on the board for corrosion/charring.

Failing that, if you have some spare RAM you can try, that would be my next suspect. RAM training can cause a system to power cycle until the cycle threshold is reached but given this is a 20min interval I doubt that is the cause. Far more likely the RAM is either defective or the wake voltage the board is providing it is insufficient to wake the RAM. Once you have had the system on for a while and the RAM physically warms up the voltage required to wake would be lower and POST will happen. If the system is not powered off at the wall the board will keep a tiny charge going through the RAM and this would not happen. It does fit with your symptoms but the power delivery would still be my first stop in troubleshooting any POST related issue.

Failing RAM causing the issue the next step is CPU, in much the same way the RAM will have a wake charge or voltage defined in the firmware, so does the CPU. In this case however if the voltage is inadequate then the CPU is defective, this spec is set by intel and not varied from one manufacturer to the next. 

At this point if nothing has helped I would contact your place of purchase and inquire about an RMA, if no longer possible then try Tech Support directly and see what they suggest for you.

Good luck and let us know how it goes