If the directions say that you should beat your cream and sugar until you have peaks, then whisk in eggs then that may be a possible cause of your results. You've just put loads of air into the cream, then you're supposed to beat eggs into it, which will punch some of the air out. What I would do is beat the eggs separately (I'd not just beat them until mixed but beat them until they start to turn pale, that adds more air), then fold them and the flour into the cream.
Be careful not to overbeat the cream as well, if you overshoot stiff peaks you'll start to turn it to butter, and then you've lost your lift. I'd stop a bit short of stiff peaks, leave it a bit looser.
Leaving that much sugar out without reducing the amount of cream would mean that you've got too much liquid in your cake. If you have too much liquid it won't be able to crystallize and the cake, while it will rise, will not be able to hold shape and will collapse. I would compensate by leaving some of the cream out. If you weigh all the dry ingredients you can work out a ratio of dry to wet ingredients (the eggs and cream separately). Then if you weigh the amount of sugar you plan to remove you can work out how much of a percentage of each wet ingredient to take out. Or you could just wing it and use 1 1/4 cups cream and 1 3/4 eggs and see how it goes, that's probably about right anyway.
Chiffon cake, like it's cousin angel food cake, is mostly air. A big pile of protein bubbles stiffened with a little starch.
One very important step is not reflected in your recipe:
When the cake is completely baked the proteins have set and the starches have gelatinized but the starches are still very soft. The cake won't have its firm structure until the starches have been able to cool and set.
All recipes call for inverting the cake immediately when it comes out of the oven. The cake is allowed to cool, inverted and still in the pan, for a good hour to ensure the starches have set.
Special pans with legs or a long tube center are used for this:
Notice that in addition to the feet that it is a tube pan. This is important because the interior of these cakes is very insulated and won't be able to cook completely before the outside is overcooked.
When my angel food cake pans were packed I had good success using a pot with a mason jar in the middle. I was surprised it came out perfectly.
TL;DR -
Use a cake pan with a tube,
Invert the cake right when it comes out of the oven and let it cool completely.
*Note
Catija is correct that these amounts look way too small for a full cake. If you are trying to bake that in a full sized cake pan it could possibly be over rising and not have enough structure to support itself when it comes out.
Best Answer
Nothing is wrong.
Continue baking until the toothpick comes out clean. The toothpick test is more definitive than time and temperature (and temperatures are not always what they claim, but don't drastically alter yours without feedback from a RELIABLE oven thermometer..)
As I mentioned in a comment on another post, I once had a bad recipe (from a fancy book, even) that I followed with great care, making chocolate soup twice in a row by following the recipe EXACTLY. A wiser than I at the time relative pointed out that there must be a problem with the recipe, and I should just bake until it was done (per toothpick) and it made cake.
A number of factors other than "oven thermostat inaccurate" can lead to times being off - shiny .vs. dark pans, temperature of the batter at the start of cooking, altitude...