Indeed. But Occam's Razor only applies to a conclusion's relation to the information you have at hand. It is conceivable that if you collect enough information the same heuristic can lead you in a different direction.
It should be able to confirm his genetic relationship to his putative great-great-great grandchildren, and thus let a lower limit on his age. That and other documentary evidence of him and his descendants could make his age seem plausible. In a world with seven billion people, outliers can be very unusual indeed.
The thing is that age isn't the result of one thing. It's the confluence of multiple systems that only evolved to keep us going until 65 or so. With modern conditions that's closer to 85, but after that all our different systems start to fail, and fail hard. You need a lot of luck (and genetics) for each one of those systems to hold up.
Lots of people make it to 90, a few to 100, some exceptional ones to 110, if you make it to 113 you might be the oldest in your country, 115 and you might be the oldest on the planet, 120 and you're the second oldest person ever. And that's if you're a woman, if you're a man you can chop about 3 years off of each of those estimates. For a man to be 120 would required extraordinary scrutiny, 125 would be absurd, 145? You're looking at about 3 or 4 layers of exceptional outliers.
To make it to 145, you'd need a subgroup with unprecedented genetic differences. This isn't Usain Bolt running 9.58 when everyone else is 9.80, or East African's making 2:10 marathons look routine. This would be a sprinter running the 100 in 8.5, or someone else running a 1:50 marathon. It's just not something that happens.