Sorry it's a long answer but it's complicated (the reasons) for such a simple thing as a flush. Yes your reasoning is almost correct but please read on.
I wish it was so simple and one of the reasons the HX machines I used to test in the old days were such a PITA to do. Yes the volume of water in the HX is the same so if we talk about exactly the same machine an increase in boiler temp causes an increase in HX thermosyphon temp. They usually (should) have injectors fitted for incoming water, this will cause some mixing and heating as it enters and the water that comes down the group pipes is not quite at thermosyphon temperature....as it enters it's being reheated in the HX and the heating elements kick in. This means after the flush the water is heating right back up again. This first effect varies a lot by machine...an Andreja might require a 30 second wait after flushing, an Izzo might be 10 seconds or pretty much flush and go by the time the PF is loaded. So boiler temperature affects this step.
The second area affected is the group and if you like, the metal in the system, the air in the case (everything that isn't water). This all serves to help maintain heat, the group temperature will have a direct effect on shot temperature. When you flush the group temperature comes down, but if you start off hotter it doesn't come down as much for an Identical flush starting from a lower temperature.
Now these effects above are subtle but will affect initial brew temperature as the water hits the coffee and, intra pour stability. With a small change in temperature (1-2C) then you wont see too much difference, but moving from 123 to 130 you might see quite a lot. The HX design in the particular machine and any thermosyphon restriction, size of thermosyphon pipes, insulation, temperature control mechanism etc.. will also affect greatly how it reacts to changing boiler temps. There are a few machines so bad that if the boiler temp is too high, they become very difficult to flush down and warm up again almost instantly. One is coming off the market as a new machine the other is still on the market and unsuspecting owners have no idea (initially), usually it takes about 12 months for the penny to drop.
The other test I do when temperature testing is an "on test". Warm up machine, after warm up period say 30 minutes (flush then check shot temp), then leave it an hour (repeat), then leave it 2 hours (repeat_. Try it you will see the flush volumes increase, surprising because you wouldn't expect any real change after 1 hour. It's a super PITA when setting up dual boiler PID settings as this and other tests have to be repeated for each temp step from 90-96...takes lots of time.
All this is why I used to put a recommended pressure/temperature for the boiler on machines HX I used to test and write manuals for or recommend to BB. With one exception, I don't test HX machines these days. The One exception is the Lelit Mara X because it's very unusual. I had a prototype back in July last year and it failed some special temperature testing as it was designed to be a "better HX" that didn't need much if any flushing. In a lab that may have been true but not in the real world used like an owner would use one. They went back to the drawing board after my report and solved the problems...In January I will get a production Mara X to side by side with the prototype and I will spend time going through the differences and the journey from prototype to final production and the case off changes.....keep an eye for it on my review site and youtube channel I think you might find it interesting.
As always things are a compromise in any machine, the HX is a larger compromise but as we compromise on pressure, grind, roast level at any specific point in the shot, as long as it all works together to produce an acceptable beverage....no problem. It's when that compromise (for any element) falls way outside the reasonable parameters we start noticing a difference. So for your specific machine I don't know the flush down volume/time, or the ideal wait period before lock and load, or the time between shots where you don't have to flush or require a reduced flush volume.. You will have to test these things yourself. You can get a pretty good idea for free by watching the water flash off to steam from the shower screen. If you pull a shot and then prep for a second, does it flash to steam again before you lock and load? If it does, how quickly does it calm down again (usually much faster) and it's easy to over flush as group temp changes as well.