Generalization of any() function with switchable default parameter for empty iterables
I really think this is a bad idea. A function called any returning True if there not only isn't anything true but even not anything at all... that's just wrong. It sure isn't a &quot;generalization&quot;. Shadowing the built-in is not a good idea, either. Maybe just call yours any_or_empty (without a default)?
For the emptiness check, the counting seems like overkill and I think a Boolean is clearer (and probably faster, due to not going through enumerate):
    empty = True

    for item in iterable:
        if item:
            return True
        empty = False

    return empty

Or with a sentinel, to get rid of the assignment in the loop:
    item = sentinel = object()

    for item in iterable:
        if item:
            return True

    return item is sentinel

For best speed, you could just handle the first item yourself and then let the built-in handle the rest:
def any(iterable, *, default=False):
    &quot;&quot;&quot;Modification of the built-in any() function.&quot;&quot;&quot;

    it = iter(iterable)
    sentinel = object()
    first = next(it, sentinel)
    if first is sentinel:
        return default
    return bool(first) or _any(it)

Your test code uses the words &quot;truey&quot; and &quot;falsey&quot;. That's not Python terminology. See the Python docs, you'll never find those words. It's true and false, as you can for example see at if, Boolean operations and Truth Value Testing.
The name falsey_iterables is ambiguous if not misleading, as it doesn't yield any false iterables. They're all true. They just don't contain anything true. I'd suggest iterables_without_trues. And iterables_with_true instead of truey_iterables for consistency.
Did you have this idea because next, min and max offer a default parameter? In their cases it makes sense, as they rather need non-empty inputs. And meaningful defaults are possible there, for example min with default=float('inf') could be a useful default for the code using the result.