What did European/American historical cooks do with the egg whites?
There were certainly uses for egg whites that didn't involve eating them:

Clearing beer and wine (using egg white as finings).  This isn't common any more, and anyway only uses one egg white to six gallons.

As an adhesive in bookbinding and gilding, and as a size.

In makeup:

Wikipedia - ancient nail polish
Royal Museums Greeenwich - as a skin treatment
also various sources say as a base for rouge, foundation and other preparations, or to conceal wrinkles.



None of these would use a lot, of course.
I suspect that most leftover whites would have been used up in cooking. Apart from the obvious meringues of various types, they can be used:

As an egg wash on pastry.
In place of whole egg before coating something in breadcrumbs
Finings are also used in clarifying stock (link is to an alternative method proposed by Heston Blumenthal).  Consommé was probably popular around that time; certainly Mrs Beeton, writing in England in the 19th century included several recipes.  While she mentioned that clarifying may be required, her consommé recipes don't call for the use of egg. Fannie Farmer's Boston Cooking-School Cook Book (1896) does explain this method, though without quantities.
Or simply added to many things that use beaten egg (I use up half eggs, either half, in an omelette with whole eggs).  In fancy households that could mean servants' food - after all there are many low-effort ways of preparing egg.  In a smaller household, everyday meals could use them up, though quickly without refrigeration

They can also be fed to many domestic animals, but given that most animals probably lived on scraps, this is effectively discarding them.