Fluting is probably the easiest way to strengthen armour without adding weight. Yet it seems like after the mid 16th century, fluting on armour was minimal.
Why did it fade out, after being so popular during the Maximillian era?

It's to my understanding that it was just terribly labor-intensive to make. It was wonderful armor in terms of a weight-to-strength ratio, but the amount of skill (and therefore money) it took to produce that type of armor made it too un-economical ( <-- technical term?) of an armor, especially once renaissance armors came about, what with their fancy-schmancy with multi-lamed segments.

I think that's at least part of the reason.
Later field armour seems to have largely been on the functional yet undecorated side.
Perhaps cuirassiers' didn't have the same status as knights had, hence their slightly simpiler armour.

The changing battlefield likely played a part in it's demise as well. Firearms were becoming more potent and more common. Why spend the time and money to make fluted armor when it's less and less effective?

Fashion, as much as anything I suspect. The style of armours changes dramatically and even during the 15th century fluted armour was very much a German thing, the Italians were doing quite different styles.

I think the role of heat treating probably played a part as well. In the time when fluted Maximillian armour was popular heat treating was practically nonexistent, and was terribly hit-and-miss when they did try it. From what I've read, by the end of the 16th century it was getting better, though it was still very imprecise and seems to have been successful more by accident than design.

probably has something to do with the relatively flat areas between the flute lines, which could probably "catch" a musket ball and let it use it's force to it's full potential

Yeah, in a book I read, The Knight and the Blast furnace, they had metalurgical results on a number of armour pieces. A piece from 1395 Italy was hardened, medium carbon steel, with a measured VPH 374.
And pieces from later dates got more consistent in their being hardened. (To a certian point, then I believe steel quality declined somewhat.)

Now that firearms are mentioned, I think that's probably why. Fluting strengenthens against sword blows, etc. by making the armour more rigid, though firearms are mostly penetration, and fluting would have probably had a negligable effect against them.

Cavalry were still going up against lances, pikes and swords regularly when Maximilian armour went out of style. I'd say its decline had more to do with fashion than function.

Two points to me come to mind-

One is that, from what I have seen, no one has ever proved fluted armour was thinner on average than the non-fluted counterparts. From the pieces I have seen this has never been the case.

Two is that there is a large amount of 16th century armour that was fluted just less and less as the century goes on. My guess is that eventually it was just not seen as really being worth the time investment to its defensive use.

RPM

It seems later 16th century armors had etching and gilding for decoration purpose. Was fluting on Gothic armor the same purpose as for those one piece breastplate with a medial ridge in terms of functionality? If the medial ridge is also fluthing.

Because fashion.

Its fashion, cost effectiveness and efficacy. There's no one reason and everyone's reason for phasing it out will be gradually different. If I had to pick one though its cost.