I agree that a solid colour on the sides of a canvas looks bad. The reason we started hanging paintings on the wall without frames (since the second half of the 20th century) was to accentuate the paint, the process, the craft, and often leaves the evolution of the painting visible. Covering it up again makes it look like a very cheap alternative to the frame.
A continuation of the scene on the sides of the painting can often be seen when an artist finds out at a later stage during the process that there is a better composition or crop available. Part of the painting gets eliminated this way, but at least remains visible.
This also emphasises the process. If a continuation is painted onto the sides after the framing, however, this will likely be visible. So not something you should (be able to) choose for, in my opinion.
I don't know what exactly you are talking about when you say "blurred background on the sides (same texture as the edges of the painting, but blurred)", but I think this will again seem like a cheap trick, and I would advice against it.
Similarly, a "mirrored reflection" is quite a vague description to me, but idem dito.
The bottom line: if you want to have the edges remain visible, stretch the painting unto a stretcher the way the artist intended, and leave the edges as they are.
If that is not to your liking, you can add a frame to the painting.