There is a concept of "Eid echad ne'eman b'issurin" which means if a person testifies on something that is questionable we believe him. An example would be regarding kosher: if I go to a friend's house and he serves me chicken, and he tells me it's kosher, I'm allowed to take his word for it. So if a restaurant owner is an observant Jew, must he have a kosher certificate?
I was told by R' Dovid Fink shlita that, speaking very generally, Ashkenazi poskim tend to require kosher certification, the reasoning being that the standards of kosher might be different from your own. Sefardi poskim, on the other hand, tend to consider a restaurant kosher if there is a visible observant Jew working there, exactly as you describe. As others have noted, though, there are many for-sure kosher restaurants that do not have independent certification that nobody would reasonably tell you to not eat at.
How do you know who the restaurant owner is or if he is trustworthy? If you meet a stranger and he invites you to his house, how do you even know that he keeps kosher. It used to be that there were stores which everyone knew were kosher because of the reputation of the owner. Nowadays, stores are owned by people who may or may not be trustworthy (or adequately knowledgeable about kashrus). It is like picking up a product with a 'plain k' on it rather than an actual copyrighted kashrus logo (such as the OU).
Nowadays, there are people who claim to 'keep kosher' but do not really know what is or is not kosher.
Even if you know the owner and trust him, he would need to show a kashrus certification so that other customers who do not know him personally can be certain.