How can I mount a TV with widely-spaced studs and weak wallboard?
That's not a single stud mount any more than the one you have is--it's just a different design for a different purpose. Use your mount with two or three good lag screws into the center stud and call it a day. 3/8" x 2-1/2" should do, piloted to 1/4" full depth plus half an inch. 

If your mount flexes a lot it's crap and you should buy one built to a higher level of quality. The mount I bought 12 years ago weighed twice what most modern ones do, but one bolt would've done the job even with the early, heavy TV we had.