.element-invisible defined in CSS

In core .element-invisible class is defined in system-behaviors.css which I have disabled just like over 50 (!!!) other core stylesheets.

While there is nothing wrong with core stylings for this class itself, system-behaviors.css contains a lot of other styles which are hard to overwrite.

I could potentially re-enable system-behaviors.css, but having to deal with all core stylsheets is a nightmare, I have already tried that in some old revisions of the theme (back when it was called Kiwi) and I found myself overwriting most of the core stylesheets anyway.

It seems like a better course of action would be to just overwrite those core styles that need to be overwritten. Disabling any core stylesheets in a core theme has a real potential to break the admin interface. Plus, when you want to a style from a core stylesheet, it won't require duplication in your own theme, which could otherwise be a problem when there's a core update.

If this stays in contrib, you can do what you want. But if this is to be a core theme, it cannot remove any core stylesheets. kat3_drx's comment is spot on.

I could eventually enable some important core stylesheets like system.css or system-behavior.css, but there is no way that I will rewrite the entire theme from scratch just to use over 50 core stylesheets from core. This would increase the size of the theme at least twice, introduce a whole lot of new bugs and make it an unmaintainable mess.

Relying on core stylesheets might be a good idea for theme that does little customization, but if you do extensive styling of all modules then you would end up overwriting most of them anyway.

Needless to say that core stylesheets are over-specific, impose outdated techniques and overall try to do to much. I'm planning to fix that in core for Drupal 8 using the work that has been done so far in Corolla as basis.

I have taken the bold step and enabled all Core stylesheets again. Furthermore I removed the contextual.js and removed all styling of the contextual links from the stylesheet. Also had to rename some stylesheets. Because if you have a system.css in your theme, the original system.css does not load anymore.
Maybe I had a non-matching version of core (Alpha 3), since neither the contextual links nor the main Menu showed up for me. So I threw out the overwriting of main_menu to main_links as well.

Attached you find the result.

The front end looks pretty much the same as in the original theme, and the back-end is a bit uglier, but by no means broken.

I'd say this is by far less work than suspected and doable until the deadline, no rewrite needed.
@Jarek: you overwrote a lot of the core styles anyway by using the same selectors, so most stuff is already done. Design fine-tuning is not the decisive factor for the readyness for core (I mean some pixels offset here, a missing border there, an ugly form over there).

Have a go at it, best make a parallel installation to be able to compare. What also helped me a lot: If I see strange effects, I quickly switch to Stark, to see if Core is causing the bug, and if not, it must be inside Corolla and can easily be solved.

Revision RC2 re-enables all core stylesheets.