In some ways you are right that you will not increase the welding problems by increasing the hardness, in that the weld will have the same composition, toughness, etc. However, the harder will make the weld possibly more difficult in other ways. The stronger material will provide higher joint restaint and place more residual stresses on the weld. There is also more chance of hydrogen cracking in the base metal due to a more susceptable microstructure.
In general, the weldability of carbon steel is considered to be good if the carbon content is held to 0.3% or less. If the material is cold worked or otherwise processed to increase the hardness, the tendency to initiate cracks is increased. Joints that involve intersecting welds can be more troublesome and prone to cracking because the forces developed by the residual stresses are added as vectors. The resulting force can exceed the tensile strength of the base metal. If the base metal is hard to begin with, i.e., has low ductility, the tendency to crack is greater than if the same base metal was not "hard". The thickness and the restrain of the joint must also be considered.
The cooling rate must be considered. Rapid cooling is going to result in harder weld deposits and heat affected zones. To minimize the potential to "harden" the weld and HAZ, preheat is employed.
A piece of steel with lower carbon content that has been cold worked or austenized and quenched can be harder and less ductile than a piece of steel with slightly higher carbon that has been annealed. How hard the steel can become is a function of the total alloy content, i.e., carbon equivalency, and the cooling rate from the austenizing temperatures, i.e., as the weld solidifies and as the weld and HAZ cool, if heat treatment is the only variable.
Best regards - Al