supposedly egg whites whip up much easier in copper bowls
Copper does indeed make egg whites beat up more quickly - there's a factual
chemical explanation for it, and fwiw, I can vouch for it from personal experience. (And if you'd rather not buy/deal with a copper bowl, I can also say from experience that the pinch of cream of tartar also works quite well.) But due to the shape, on the one hand, and the reactivity/potential toxicity of the material on the other, they're basically limited to hand-beating, and pretty much define the word "unitask," so might not be worth the cost/storage space unless you have a fairly frequent use for them. I guess you could use the largest bowl for cooking small amounts of sugar/candy, but the shape would make that problematic on a home stove. (It should go without saying that you wouldn't want to have to hold the bowl by hand while cooking sugar to temps significantly above 212F.

)
And while that
is a good price even for the Old Dutch brand, do be aware that they're pretty thin (compared to the much more expensive "big" copper brands), so get banged up pretty easily, but that's only a matter of appearance of course, not functionality. Lastly, Home Depot (of all places) happens to have the
4.5qt bowl available online for $32.95, with their usual free shipping to store, if anyone would rather deal with them than Groupon. (The 5qt bowl and and 3 piece set aren't discounted there though, so are outrageously overpriced for the quality.)
are egg whites the only reason to have copper mixing bowls?
Yeah, more or less. Copper is highly reactive, which will/can affect the color and flavor of many foods - particularly acidic ones and dairy products - and for that matter is also potentially toxic in the chronic/long-term sense, so aside from its physical characteristics, its use in the kitchen is rather limited. Pretty much the only traditional uses for unlined copper are beating egg whites and cooking sugar (for caramel) or candymaking (like fudge, taffy, etc.) As you'll also see from the Groupon listing, (equally unitasking) zabaglione pots have also traditionally been made of unlined copper. And of course a round bowl is a PITA for mixing anything heavy when you'd want to be able to set it on a surface to support the weight. (For egg whites, one generally just holds it one hand and the whisk in the other.)