If I wouldn't want to use refined white sugar in recipes warranting sugar and eggs beaten fluffy, what would be the best alternative to use? Would the result be very different?
Use maple sugar.
Things will taste better*, but cost more.
*unless, of course, you are a person who does not like real maple (of whom there are a surprising number, though I cannot claim to understand that.)
For ice cream/sorbet/gelato/frozen desserts, most "sugar substitutes" (low calorie sweeteners, etc) don't really work, as sugar plays an important structural role in how ice cream turns out. Being sugar, maple sugar (or other "unrefined/less refined" sugars) work fine.
I find that the chemical and mechanical reactions between egg-white, egg-yolk and sugar is quite an delicate and complex one. I find that there is even a difference in using white refined cane sugar and white refined beet sugars when baking and the amount of effort of creaming eggs and sugar. Sugar substitutes that claim that they "behave" the same as sugar also don't seem to react in a manner that you'd expect.
I would suggest that you try a range of sugars... The worse of them seems to be corn based sugars. The eggs just don't behave at all...