I'd recommend trying it. It takes a few tries to get the correct input parameters, and I've noticed anything approaching 4× scale tends to add unwanted hallucinations.
For example, I had a picture of a bear I made with Midjourney. At a scale of 2×, it looked great. At a scale of 4×, it adds bear faces into the fur. It also tends to turn human faces into completely different people if they start too small.
When it works, though, it really works. The detail it adds can be incredibly realistic.
That magnific.ai thingy is taking a lot of liberty on the images, and denaturing it.
Their example with the cake is the most obvious. To me, the original image shows a delicious cake, and the modified one shows a cake that I would rather not eat...
Every single one of their before & after photos looks worse in the after.
The cartoons & illustrations lose all of their gradations in feeling & tone with every outline a harsh edge. The landscapes lose any sense of lushness and atmosphere, instead taking a high-clarity HDR look. Faces have blemishes inserted the original actor never had. Fruit is replaced with wax imitation.
As an artist, I would never run any of my art through anything like this.
Look for SuperResolution. These models will typically come as a GAN, Normalizing Flow (or Score, NODE), or more recently Diffusion (or SNODE) (or some combination!). The one you want will depend on your computational resources, how lossy you are willing to be, and your image domain (if you're unwilling to tune). Real time (>60fps) is typically going to be a GAN or flow.
Make sure to test the models before you deploy. Nothing will be lossless doing superresolution but flows can get you lossless in compression.
I haven't explored the current SOTA recently, but super-resolution has been pretty good for a lot of tasks for few years at least. Probably just start with hugging-face [0] and try a few out, especially diffusion-based models.
This is called super resolution (SR). 2x SR is pretty safe and easy (so every pixel in becomes 2x2 out, in your example 800x600->1600x1200). Higher scalings are a lot harder and prone to hallucination, weird texturing, etc.