Mac Mini (Mid-2011) does not provide any signal after nvram/pram reset

Unfortunately, the PRAM reset is only good for CMOS/Clock batteries/capacitors under 4 years old. 2011 is much older, and could indicate a capactor that needs work on the logicboard. A 2011 Mac Mini can be had on eBay for $45. You could swap your storage into it for less than the work of fixing the logicboard, if you can find one that works from a reputable vendor.

Reason to have a 2011 Mac Mini, is compatibility with Mac OS X 10.7.5, which very few applications or printers require and compatibility up to High Sierra. It may be one of the last ones to support Firewire, but that's able to be adapted to Thunderbolt 2 for all but booting.

2012 Macs starting in October came with Mountain Lion.

2015 Macs starting in October came with El Capitan.

Mountain Lion compatible Macs all support at least El Capitan regardless.

Even 2017 Macs support HIgh Sierra. 2016 Macs support Sierra. Sierra is the first Mac OS to support the current multi-factor authentication.

Is there really any specific purpose for you to hold onto the old Mac Mini, other than possible storage?

There is one thing you could do. And that's get a tested logicboard and power supply. Or even another Mac Mini the same age, But you would want an authorized service center to help with installing the hard drive so you don't mess up the install. Just make sure it is the same age as your current computer. The storage likely is not affected by this problem since what you describe is either related to the graphics chip or power supply of the computer. That's why I think a whole computer is a better venture.