I have Gathering and Cooking for skills.  They both were ready to be upgraded at about the same time.

When I went to upgrade Gathering, there is a typo where the first option says "Leave" but it's actually what you click to upgrade the skill.

As for Cooking, it is not letting me upgrade it at all in either town.  I click the option to upgrade and nothing happens at either Cooking Robot in either starting town, and my skill is maxed where it needs to be upgraded.

I can confirm this. The upgrade choice for cooking does not upgrade the skill.

Thank you

The same thing happens if you try it with harvesting.

Say you must be lvl 5 to upgrade, but it must be possible for lvl 4 to upgrade to lvl 5 first.