> Why can't an employee from some Big-N company make something for fun on the company domain?

Because it's under the company domain?

Anything carrying its brand is linked to the company. You can't be a big company and an humble startup at the same time. If Google wanted to provide an opportunity for its employees to launch funny products without pressure they would give them a separate domain. As is, they do it this way because they think it's good marketing. That people expect big company support from big company websites go with the territory.