WASHINGTON—President Donald Trump and the GOP-controlled Congress, who opened their first year in full control of Washington on rocky terms, are closing it with a flush of late legislative achievements: a sweeping tax overhaul, a long-sought repeal of a pillar of the Affordable Care Act and a surprise deal to open up Arctic drilling—all signed into law Friday.

After the high-profile collapse of a health-care bill to replace Obamacare, the tax legislation itself is a victory that changes the narrative of the first year, some...