I understand your feeling - the back of my flat screen feels very flexible and that is the part with the mounting holes with threads for bolts. I put two small pieces of metal under the base of the TV and they were bolted to the metal frame that supported the heavy old TV. These pieces of metal are about six inches long and about one inch wide. I put a quarter inch 90 degree bend in the ends that go under the TV. The metal is painted black and if you don't know they are there you don't notice them. I did also use the mounting holes on the back of the TV. Anyhow some of the TV weight rests on these two pieces of metal which should help relieve some of the strain on the flimsy back of the TV.
Bob, Sandi & Marmaduke the Big Pug
SW OREGON 2004 Journey 39K, 330 Cat
If towing: a 2006 Mini Cooper or 1995 Wrangler