Discrepencies between std::lower_bound and std::se

The C++ draft says about std::lower_bound:

§ 25.4.3.1 lower_bound [lower.bound]  
template<class ForwardIterator, class T>  
ForwardIterator lower_bound(ForwardIterator first, 
                            ForwardIterator last, 
                            const T& value);  
template<class ForwardIterator, class T, class Compare>  
ForwardIterator lower_bound(ForwardIterator first, 
                            ForwardIterator last, 
                            const T& value, 
                            Compare comp);  

Requires: The elements e of [first,last) shall be partitioned with respect 
    to the expression e < value or comp(e, value).
Returns: The furthermost iterator i in the range [first,last] such that for 
    any iterator j in the range [first,i) the following corresponding 
    conditions hold: *j < value or comp(*j, value) != false.
Complexity: At most log2(last − first) + O(1) comparisons.

Note that this allows (by implication) one to compare a different (but comparable) type than *ForwardIterator would return, but there's no complexity limitation on the number of iterator advancements. (For a node based container, this would be O(last − first) iterator advancements.)

§ 23.4.6.1
class set { 
   ...
    iterator lower_bound(const key_type& x);
    const_iterator lower_bound(const key_type& x) const;
   ...
}

The standard doesn't detail these functions, but it's implied that these are for O(log2(last − first)) comparisons and advancements, but require the search key to be the same as the contained type.

So my questions are:
(1) Is there a way to get the speed of std::set::lower_bound and the flexibility of search type of std::lower_bound?
(2) Why isn't std::lower_bound required to be specialized for std::set::iterator?
(3) Are there implementations where std::lower_bound is specialized for std::set::iterator, or is there a reason why not?

I was hoping to find something like:

template< class Key, class Comp, class Alloc, class Lookup>  
std::set<Key, Compare, Alloc>::const_iterator 
    lower_bound(
        std::set<Key, Comp, Alloc>::const_iterator begin, 
        std::set<Key, Comp, Alloc>::const_iterator end, 
        const Lookup& find,
        Compare comp);
template< class Key, class Comp, class Alloc, class Lookup>  
std::set<Key, Compare, Alloc>::iterator 
    lower_bound(
        std::set<Key, Comp, Alloc>::iterator begin, 
        std::set<Key, Comp, Alloc>::iterator end, 
        const Lookup& find,
        Compare comp);

or:

template < class Key, class Compare = less<Key>, class Allocator = allocator<Key> >
class set {
   ...
    template<class Lookup>
    iterator lower_bound(const Lookup& x);
    template<class Lookup>
    const_iterator lower_bound(const Lookup& x) const;
   ...
}

But I doubt it exists. Obviously, all of this can be expanded to other tree-based containers, and other algorithms. I'd code it myself, but it would require me to use implementation specific code. This idea came from this question: Efective search in set with non-key type

A std::set container is ordered according to a comparison object that is given at construction time. When std::lower_bound is called, there is no way to check that it was passed a matching comparison object, so the implementation can't know whether to use the standard algorithm or one specialized to sets, as the latter is only valid when using the comparison object used for the set's ordering (or one which gives the same results).

The two example prototypes you added won't work:

Specializing std::lower_bound to work on std::set iterators:

This won't work for the reason given above: There is no way to check if the given comparison object matches the one given in the set's constructor. Your prototype only checks that the type of the comparison object matches, but there can be different comparison objects of the same type.
Making std::set::lower_bound take a template argument:

This may make it incompatible with the set's comparison object, since that object's operator() will usually not be templated, and expects only arguments of type T.