Do horizontal stabilisers provide any lift?
The horizontal stabilizer provides lift, but usually in the negative direction.
Could one provide positive lift?

The answer depends on cg location.

Forward CG, the answer is no.

Aft CG, the answer is yes but only if the CG was aft of the center of lift.

In other words, the more forward the CG is compared to the center of lift, the more downforce the tail must provide.  The lift from the horizontal stab in the "wrong direction" must be countered by the wing, and thus there is more induced drag.  For a given aircraft gross weight and thrust/power, there will be a higher cruise speed with a more aft CG.

There are practical limits to extreme fore and aft CG positions, and so each aircraft has an allowable "envelope" for loading for safety margin.

Aircraft are designed to have a range of allowable CG locations to make loading easier before the flight (especially for cargo aircraft), and also to allow for passengers &amp; crew to move about the cabin in flight.

Horizontal stabilizers could be sized smaller but that would decrease loading flexibility, decrease passenger comfort, and decrease aircraft handling performance; often all airfoils and control surfaces are designed with many factors in mind, for good control harmony, good ride in rough air, for stability on instrument approaches, for flexible loadings of various belly and main deck cargo holds / passenger configurations, etc.

Flying wings and delta wing designs come to mind as aircraft without horizontal stabilizers, such as the Northrup Spirit or Convair Delta Dart.

Your other question, would manufacturers design horizontal stabilizers to provide lift?
Hopefully you can see by now this is not a design goal.  Wing design is so advanced that designers are making amazingly beautiful and high-performance wings that are most efficient at creating lift.  Look at a Gulfstream V wing or Boeing 747 wing.  It is not efficient to create a wing that doesn't work great and then needs an additional surface to help make more lift.  Exceptions to that might be bi-planes or tri-planes, as they used several shorter wings instead of 1 longer wing, but that is not practical in air transport.  How many Boeing Biplanes and Airbus Triplanes do you see traveling the globe? It's simply better to put all the lift needed into a single wing design.