I still don't fully understand how ground and chassis connection differs
That is partly because people use the term "ground" very imprecisely, lots of connections with different functions get lumped into that term just because they have a DC connection.
"The ground" is dirt outside your house. There is a big metal rod pounded into "the ground" that the power line connects to as part of the design to not direct lightning into your house along the power wired.
Chassis is a metal box which shields the internal circuitry from RF interference (nearby radio and tv stations, mostly). Depending on the design of the power supply, the equipment chassis is often connected to a safety wire in the power wiring which also connects to that power line ground wire at the breaker and power distribution box. The connection back to the actual planet earth is somewhat coincidental, a wire running through the walls that is many yards/meters long does nothing at radio frequencies, so is not actually involved in the RF interference protection function of the chassis at all.
The reference connection for the circuitry is the point used explicitly as the voltage measurement reference point. Because that part of the circuitry connects back to the chassis at some point it gets called "ground" as well, even though it has nothing to do with lightning protection and so does not need any connection to the earth for the circuitry to function.
One thing to keep in your conscious thoughts is that even though that circuit node is called "the" reference for the circuit, it is made of physical materials that have resistance and inductance, and Ohm's Law applies just like any other conductor. Any current flowing through "the" ground will cause a voltage difference along that current path, so one point of "the" ground will be at a different voltage referenced to another point on "the" ground.
You
have to be aware of where current is flowing, and what effect that has on circuit performance.
Now, back to the chassis: wires used to connect different pieces of equipment together are just another conductor in an RF field, so radio waves will generate currents on the shield connections just like it was an antenna sticking up on your car. You don't want that current flowing along the audio circuitry, which is why cable shields need to connect directly to the chassis, you want to keep all the RF energy on the outside of your metal box, and not running through your circuit.
What about when wiring to an TRS connector?
When a TRS connector is used for balanced audio connections the convention is tip is hot /positive polarity signal, equivalent to (modern convention) pin 2 on XLR. The ring is cold/negative polarity signal, equivalent to pin 3 on XLR. The sleeve is the cable shield connection, like XLR pin 1 it should be tied directly to chassis. That is actually easier to do with a metal TRS connector, just don't put an insulating washer under the nut when you screw it down to the chassis.
The ground connection (even though the transformer is unshielded)
Whoa! Transformer output windings do not have a "ground," there is signal and signal return. The entire point of a transformer is that the windings are galvanically isolated, so there is no need to connect either end to the chassis or any other circuit node. You choose the connection points based on where you want the current to flow, which ideally is out one end of the winding, to the input circuit of the receiving piece of equipment, then back to the other end of the transformer winding to complete the current loop.
For an output transformer the output winding connects to pins 2 and 3 of an XLR, or tip and ring of a TRS connector. That is all, just direct to the connector.
Since you're converting to a balanced connection, there is no connection to the ring (R) of the TRS connector.
The "balance" in a balanced connection is the impedance of pin 2 to the chassis and of pin 3 to the chassis, or of tip to chassis and ring to chassis. If you connect one side to the sleeve then you have completely
unbalanced the connection, one side has high impedance to chassis and one side is a direct short to chassis, which is the opposite of what you said you are attempting to do.
