No, a T-stat is always needed to regulate coolant temperature.
The T-stat is widely misunderstood. It simply adjusts its opening to regulate the temperature at its bi-metal spring. For some reason, many mechanics seem to think it continually opens and closes as the engine runs. Perhaps they are thinking of a home thermostat which cycles your home heater on and off. That is termed "on-off" control or "bang-bang" in older textbooks. A car T-stat is a "proportional controller", using "negative feedback". My guess is that it reaches a steady-state opening position, rather than constantly oscillating open and closed, though I have no videos. The later is termed "feedback instability" and can occur in a proportional controller which has too much gain (too touchy, not desired).
One property of a proportional controller is that it can't regulate to an exact setpoint, except at one operating condition. If the heat load increases, the T-stat must open further to allow more coolant flow, in an attempt to hold the temperature setpoint. But, the only way it will open further is if its sensing spring gets hotter. Thus, it winds up at a higher steady-state temperature under a higher heat load (termed "proportional droop"). That is why you see the dash gage creep up when under load, like going up a steep mountain grade or on a very hot day. Of course, once it is fully open, it can no longer regulate the temperature. It is of course wrong to think that installing a 180 F T-stat instead of 195 F will keep your engine from overheating at say 220 F, since the 195 F T-stat should also be full-open long before then. Another reason for the dash temperature to vary is that its sensor is sometimes in a different place than where the T-stat regulates and coolant temperature varies around the engine.
There is a way to avoid the setpoint offset with feedback controller, but requires "integral action", which I don't think any simple T-stats have. Some have extra parts for "flow balance", intended to make flow forces have less effect on the opening. Higher gain lessens offset, but as-mentioned, too much gain can cause oscillation. Anyone interested can google the terms above, or peruse an engineering textbook on automated control.
If anyone has fooled with Euro engines, you will find that most have an "alien" 2-headed T-stat. One disk regulates radiator flow, as in U.S. engines and the outer one closes off the bypass circuit as the engine warms. U.S. engines, at least classics, have a fixed bypass path which never closes off, even though doing so is likely better once heated up. I think the reason for the bypass is to avoid dead-heading the water pump when the T-stat is closed (higher pressures, risk of cavitating at high rpm).
