"I eat cake." should be accepted as another option as well, or am I wrong?
yeah, it should be
I am eating a cake, should also be accepted
That sounds like eating a whole cake, which would be "(Minä) syön kakun."
"I am eating" specifies that you're in the middle of the action, not that you're talking about the action as a whole, so kakkua should work.
Syön kakun would be more like "I'll eat a cake."
There is no context to justify Continuous tense. "I eat cake" is also correct.
How funny that I've lived nearly 50 years of Finnish family conversations and I only just found out it's not 'syän'.
To be fair, in some dialects it is "syän" :) And in some other dialects "syän" means a heart - "sydän" with the d dropped off.
Syämme syämia ...