It would seem to be that this establishment is not to be patronized and or trusted to be serving kosher food unless they have a kosher certification.
This is based on a rule enacted by the Vaad Ha'arba Artzot - The Council of the Four Lands, the central body of Jewish authority in Poland from 1580 to 1764. The Vaad
instituted that it is forbidden to eat any food or drink any wine from anyone, even someone who is assumed to be trustworthy, unless the person has a letter from a head of a Rabbinical Court testifying to the Kashruth of the food.
הדרכי תשובה יו"ד סי' קי"ט סע' א' אות ו', הלחם הפנים, ועוד אחרונים:
תקנת ועד ארבע ארצות שלא לאכול שום מאכל או לשתות ײן מאדם אחר אפילו ממי
שהוא מוחזק בכשרות אלא א״כ יש בידו כתב הכשר מאיזה רב אב״ד שנעשה בהכשר
The predominant opinion is to assume that this enactment only applies to establishments, but a private person can be generally trusted. Although there is a concept in Halacha called עד אחד נאמן באיסורין - that generally people who are trustworthy are believed to verify the kashrus of food, nonetheless the Vaad made this enactment. This is the source for modern day Kashruth Certifications. However, there is a debate if this enactment was for only their times, or if this was an enactment for all generations. However, in the Sefer "B'nesivos Hakashrus" the author writes Vaad did mean in for all generations.
Thus it would appear to be that nowadays if a restaurant were to operate without a certificate attesting to its kasruth from the Beth Din, despite the fact they claim the food is kosher,nonetheless it would seem to be prohibited to procure food that establishment.
Ultimatly this is how the Author cited above concludes:
אבל מוכר שאינו מכיר את המורכבות והמסובכות בכשרות שבזמנינו ואינו יודע
מספיק ממה .צריך להזהר, אע״פ שהוא מוחזק בכשרות אין לקנות אצלו כמבואר
חילוקי הדינים בפנים.
However, one who is not well versed in the intricacies of the Laws of Kashruth in our times, and does not what of what to be concerned for, even though he is a trustworthy person, nonetheless one should not acquire food from him.