Your last post with description of the classic solenoid “click” sound with failure to crank sounds like classic bad battery symptom...or bad battery ground cable connection. Don't forget the engine ground braid connection to the body through which the starter current flows...but unlikely if you haven't messed with it lately. Mine was heavily corroded and my starter worked stronger after cleaning and tightening....but again, not as likely.
I know you said the battery was new and see comments where you have measured "12V" before, but do check that it is closer to 12.6 volts which is where I'd expect a brand new, fully charged battery to be with ignition and lights off. If its measuring only 12.3 or less at its main terminals, either it isn't fully charged, it has an internal defect, or there is some thing drawing a fair bit of current that shouldn’t be there with everything turned off.
It’s not that unusual to have a brand new battery that measures good voltage at the store, without a load but then just cant supply much current due to high internal resistance/defect. Another thing to check is to watch the voltage at fuse #1 while you are trying to crank the engine. It should droop somewhat below 12V while in the crank position but certainly not drop to zero. If it does drop very low then you must have a bad battery or cabling from the battery. You eliminated the starter as a suspect.
The voltage regulator, good or bad, shouldnt have had any effect on starter function. Although it could possibly have an internal short that drains the battery, but seems unlikely.
Sorry if you've been over some of this, just really curious about this issue and wish you luck!