Honestly the old logo is four colors visually, you brain is shortcutting the overlaps because it understands them.
Aside from physical constraints, I think there is a general argument to be made for simplicity. If something is more complex, there should be a reason for it. Otherwise, toss out the complexity. What is the reason for having more colors? There are other ways to convey overlapping lines, and I would argue that some of them are less complex, though it would be a lengthy argument -- it isn't exactly obvious how the complexity of an additional shape can be measured against the complexity of an additional color, and you can ultimately just subjectively weight one of those factors heavier than other and come up with a different answer. But I think the general case for fewer colors being less complex is pretty sound.
[Edit for grammar]
You don't _ever_ want people to be re-producing your logo (and especially free-handing it!). That is why they pretty much all have a "brand" page to pull assets from: https://brandfolder.com/slack/logos.