Apple has done a great job of marketing the photo capabilities of the iPhone, but the fact remains that a larger lens and larger sensor will produce a better result, more "substance" to the image.

True, true. There are other factors as well which make a true "camera" picture "better" (in technical terms rather than artistically) than one from a phone, it is just the physics of capturing light. To be fair, camera vs phone all depends on what you want to do with your pics whether a phone is good enough or not. Also, the conditions under which you shoot. Low lighting, telephoto, or "action" shots will clearly show the deficits of a cell phone. However, for the first time I tried my cell as my back-up camera when I went to Europe this year. Obviously I was glad I had my full-frame camera for (to me) "important" shots, but it was fine when I went out to dinner or snapped a quick pic when I hopped off the bus.

As to the OP, there has been some good advice about trying to salvage your low res pictures, so I won't repeat it. But I will add there that when upsampling there are several algorithms, so if you have a program which allows you to try different ones, do so and one may produce better results than the others. Also, as stated jpeg is a lossy compression and resaving a photo will degrade the image, however rather than a png format, I would save the new ones as a tif, which (IMHO) has some advantages over png. Don't forget, after upsampling and editing, you will have to reconvert the tif to jpeg. Here is some info on tif files, editing them and converting them. Good luck. I have also failed to notice I was not taking the "quality" of photos I wanted and it is a disheartening experience.