Plausibility: greater mathematicians than Fermat worked on this and couldn't prove it. If a 17th-century proof (so to speak) were feasible, surely Euler or Gauss - to name but two later mathematical geniuses of the highest rank - would have found it. It took 350 years or so to crack because we needed a lot of tools not available to Fermat or his successors. Many of them contributed to what culminated with Wiles/Taylor. That's not an irrefutable argument, but short of finding missing notes from Fermat, it's likely the best you'll get.

Plausibility: greater mathematicians than Fermat worked on this and couldn't prove it. If a 17th-century proof (so to speak) were feasible, surely Euler or Gauss - to name but two later mathematical geniuses of the highest rank - would have found it. It took 350 years or so to crack because we needed a lot of tools not available to Fermat or his successors. Many of them contributed to what culminated with Wiles/Taylor. That's not an irrefutable argument, but short of finding missing notes from Fermat, it's likely the best you'll get.