Which president had the most lasting impact on the major Progressive issues?
I assume that you are asking about the presidents of the Progressive Era (not about later presidents who did things that the Progressives would have liked). Of these presidents, I would argue that Theodore Roosevelt had the most lasting impact. This is because it was during his time that the first major steps towards Progressivism were taken.
TR was the president who first moved away from having the government constantly back employers against unions. He intervened in a more even-handed way in the Pennsylvania coal strike of 1902 and then got Congress to create the Department of Commerce and Labor to help regulate companies. He pushed for antitrust laws and enforced them. He pushed for the Meat Inspection Act and the Pure Food and Drug Act.
By doing all of these things, TR took the first major steps towards enacting Progressive ideas. This is why he had the greatest impact.