No I want to go farther back than 1972 and Nixon. Follow me back below the squiggle of time.
Let's go back to 1774 and George Washington. Previously as a surveyor in the employ Lord Fairfax of Virginia the young Washington had traveled up the Potomac River and on into the Ohio River Valley. He realized that the greatest obstacle to development was poor internal travel. The Potomac was the most direct route but it's many natural obstacles made river travel impractical. So in 1774 Washington introduced a bill into Virginia's House of Burgesses to build canals around the Potomac's five worst obstacles. Maryland, which shared jurisdiction over the river, rejected the plan.
Jump ahead to 1785 and General Washington is now a war hero. The Potowmack Canal Company was created and Washington was chosen as its first president. He was a frequent visitor during construction but didn't live to see it completed. He died in 1799 while the Potowmack Canal was completed in 1802 and began operation. In the 1820s the rights of the Potowmack Canal Company were transferred to the new Chesapeake and Ohio Canal Company which planned to build a continuous canal that would link the Chesapeake Bay to the Ohio River. That plan was ultimately unsuccessful and the canal stopped operations in 1924. The remnants of the canal still exists as the Chesapeake and Ohio Canal National Historic Park.
The way a canal works is to raise and lower boats with the use of locks that have a set of doors or gates at each end. The boat is moved into the lock then the gates are shut and the water inside the lock is either raised or lower via gravity, no pumps needed. As the water raises or lowers so goes the boat.
Still with me? We're almost there. Inside of Washington D.C. The Canal used the lower part of Rock Creek to connect it to the Potomac River. Across the mouth of Rock Creek was built a set of gates to control the water level in the creek. In 1963 a hotel complex was built next to the remains of the Rock Creek water gates. The complex was called The Watergate.
While some might argue that "-gate" suffix is derived from the name "Watergate" I think of it differently. At the beginning of this diary you'll see a picture I took a few years ago showing the mouth of Rock Creek. In the far background is the Theodore Roosevelt Bridge spanning the Potomac. In the center of the picture you can see the remains of the Rock Creek water gate. This is the actual "gate" from which all others in this diary are derived. Instead of "-gate" being a nebulas term I like to think of it as being a real thing you can actually go and touch if you want to.
Update: And as was pointed out to me. It's all really George Washington's fault.