Technically you aren't making bigger pixels, just putting space between the existing pixels and letting the software take it's best guess from what is adjacent to the spaces - aka interpolation. What programs like Genuine Fractals/ON1 Perfect Resize and others do is adjust contrast at edges and provide good antialiasing so linear elements that are not perfectly vertical or horizontal, or curved elements appear without the "stair-stepping" effect that comes when you do extreme resizing. Resizing software also increases high frequency contrast at transitions and removes artifacts and noise - providing a "cleaner", "sharper" appearance.

That being said, for modest resizing upwards, the resizing algorithms in Photoshop and Lightroom are pretty good. Photoshop gives you a choice as to which interpolation you can use, Lightroom doesn't.

One thing that the software will NOT do is add detail. If you didn't capture it applying resizing software isn't magically going to put it back. Any image with a lot of texture (small frequency detail) just looks weird when resized with any technique.

Also, if you are sending your work out to a commercial lab, their RIP (Raster Image Processor) will do the same enhancements on a modest up-res without needing to process the image.

The number of ppi you need when you make your prints bigger decreases as you increase the print size because it is presumed that your viewing distance increases as well. A 4"x6" print will need around 360 ppi to look sharp, while a 40x60 only needs around 32 ppi for acceptable results, and anything higher would put it in the excellent category.

http://www.photokaboom.com/photography/learn/printing/resolution/1_which_resolution_print_size_viewing_distance.htm

One last thing - if you are printing your own work, Qimage software will optimize the image for printing, doing a lot of the same enhancements a commercial RIP will do.

This article compares a few resizing programs, including Photoshop CC with Preserve Details enabled.

https://photographylife.com/how-to-increase-the-resolution-of-an-image

In any case, I usually get more controllable results by adjusting the microcontrast and applying the correct amount of sharpening and de-noising in an image. It's not uncommon for an image to require different amounts of enhancement in different areas.

But don't expect a cellphone image resized to higher mp to look like it was taken with a Phase One IQ4 150MP medium format camera. It won't.

Presumed isn't much different than assumed. I have never been anywhere that most people don't stick their noses to the image unless there is a barrier to keep them away. A few weeks ago I went to an exhibit at the Peninsula Gallery in Lewes. I got their early as I knew one of the exhibitors and her grandmother (a good friend) kind of insisted. I got to take a nice look at everything displayed as you suggest. I had already seen them on my computer.

Once people started coming in nobody stood far enough back that they could see the entirety of the images without actually turning their heads. I thought, if only Gene were here he might give these folks a lesson.

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CaptureOne is the best of all that I've tried, including ON1's special purpose application. It wasn't even close.

But I can't comment on the Adobe programs because I have not compared them to CaptureOne.

In CaptureOne you set the DPI and the size of the intended print when you export to TIFF of JPG. Inherent in that is, potentially, up-sizing and the necessary computations at the pixel level. I have posted examples here before.

I have used it to make a 4' x 6' print from my crop-frame Sony a6500 with excellent results. That's feet, not inches.