supplying fuel and air in the right measurements, would the expanding gasses exiting the combustion chambers produce any thrust?
Exposed to the open air, no. Essentially you've built yourself a fireplace. The circulation would be at best be cooler denser air being drawn in from the bottom. Hotter, less dense air would rise. Adding fuel in the "right proportions" you have ... a Bunsen burner.
The key is raising pressure in the combustion chamber to increase both mass and velocity of exiting gasses in order to generate meaningful thrust:
$ mass × velocity/second$
Thrust is a Force = mv/s = ma = kg × meters/second$^2$
Back to Heros Engine. This is a steam engine. Pressure is raised by using an external heat source to expand liquid water into steam. The expansion ratio is 1 to 1700.
Heros engine spins by expelling steam vapor out its nozzles at high velocity. But significant pressure is required to do this. Action - Reaction. (Tea kettles won't fly, but steam boilers can explode).
Back to our jet engine, along with its rocket and reciprocating (piston) relatives. These are internal combustion engines, which provide thrust from a heat source inside a "combustion chamber", creating pressure.
One little problem is that generating extremely high pressures and expelling high velocity gasses can be terribly inefficient compared to converting pressure into mechanical energy to turn a wheel, or propeller.
the quantity of fuel required to produce a given amount of thrust for a given amount of time is call Thrust Specific Fuel Consumption
It's inverse is ... Isp or Specific Impulse.
Rockets are the worst, but will function in any environment. So will Hero's engine. If you are ever stuck 10 million miles from earth with a nuclear reactor and happen to find an icey asteroid, you're going home!
Reciprocating engines create cylinder pressure in power pulses to generate mechanical energy, jets create power continuously to generate mechanical energy by spinning a turbine.
Jets have Horsepower!
Jet need their "Horsepower" to increase incoming air pressure in the combustion chamber by turning a compressor. This is called a "turbojet". In spite of being less efficient, turbojets are able to generate thrust at speeds that mechanical devices such as propellers cannot match.
unfortunately, these days, fuel is expensive
So now jet engines use their "Horsepower" to turn not only compressors, but also their fans, which are able to generate thrust at higher speeds than propellers.