I don't think so. We can very plainly feel when the hot water "runs out". It's a constant temperature and then womp, it gets cold very quickly.
Obviously there is a thermocline in the tank that rises and when the last of the hot water (presumably the full 175L) gets used, you hit the cold water layer.
We have low flow shower heads and I in particular am very frugal with my shower.
Yes, but the previous gas burner was capable of putting out 4-10kW+ of equivalent thermal power. As soon as that temperature drops below ~50C, the burner fires up and won't stop until people have stopped using water.
Its different with the new tank, because its only heating during some portion of the day, so if you are using it outside of that time zone, you will be limited to the ~300L in the tank. Whereas before, 175L was never a true limit.
Also, the new tank is using even more (10-11kW vs 9kW you calculated with gas). Either thats some measurement anomaly or the hot water use has gone up a little bit (from ~220L to ~245L per day).
edit: 220L is top end estimate though, would be a bit lower due to heat loss in the pipes.