This is (almost) strictly a user-to-user support site, and though very rarely an Apple staffer may post helpful information, it is not as a formal contribution from Apple, but merely as another user trying to offer some assistance.
Unfortunately in your case, while there have been some reports of similar behavior in other Mac minis, there has not been any clear solution found - which is why I suspect you haven't had any helpful responses. The SMC reset procedure has been suggested for this in the past, and has occasionally worked, as it appeared to do for you that one time, but generally it appears not to have helped. Indeed, the fact it didn't resolve the problem in your case after that first time suggests that it may only have been coincidental that it seemed to have worked.
There are really only a couple of possible causes as I see it. Faulty memory can cause the sort of failure to boot you describe here, and could also be complicit in wake from sleep issues too, and it's possible an external device could be causing problems too. You might try and boot with nothing connected to see if that helps, but in the broadest sense this sounds to me like a hardware failure. Macs (not just Mac mini models) do occasionally suffer wake from sleep issues, often the result of peripheral devices, but this would not be expected to also affect system boot up. Indeed, should
never do so.
Typically, if done correctly, an SMC reset either works first time or not. Doing it multiple times means it isn't the solution - as you have found. It may be that your system is suffering a dead internal battery (some Mac models simply fail to boot with symptoms similar to yours) with then internal battery fails, but all in all I think Apple need to look at this problem and try and resolve it, because even if there appears to be a solution in some user-initiated procedure such as an SMC reset, unless the actual fault is known, there can't be much confidence the problem will not return at some point.
Please let us know what Apple say - I suspect they'll have you run through the usual array of troubleshooting tests, and then want to start replacing components such as the logic board, but it will be interesting to know what they decide.