Yes, a relay would have been a much better analogy than a transformer.
I think that should be true, upon confirming that the resistance of the photo detector in the new LED opto-isolator gets below the value that it needs to activate the triac. What I don't know is what that value is, and if the substitute component will be sufficient.
Honestly, thinking through this more, without a true RMS reading AC voltmeter, and without some guide as to what the potentiometers R7 and R2 do, I am not really sure I know what it will take to confirm that the lamp is operating at 87v RMS. It may turn out that somehow an incandescent bulb will have to be put in place of the 87v lamp by means of some test jig cabling. This, to avoid destroying an expensive projection lamp. Instead use a 300W incandescent bulb . Bulb that big will be a monster. GE has one that is not a 3-way bulb. It will be mighty bright and mighty hot! And, I don't know enough about bulbs or the circuit to know if the GE bulb will sit in the circuit the same as the 87v lamp - if the resistance is different does that change the operation of the circuit? It may be that one has to take a "ham radio" approach by doing something like wiring 10 330 ohms resistors each 25 watts, and putting into an empty paint can filled with oil to dissipate the heat.
If you or someone else has some experiences to share with trying a part substitution due to part obsolescence, that may help determine the need to "realign" the power supply control board to get the lamp into its operating range.