There are at least two things wrong with this statement.
First, "the ones for which Cantor’s diagonal argument applies" is a bit vague. I assume it's supposed to be a reference to uncountable sets, but as written, it's (probably) referring to the general version of the argument that shows that the powerset of a set is strictly larger than the set itself. Thus, a set "for which Cantor's diagonal argument" applies is a powerset.
But not all uncountable sets (or even all uncountable total orders) are powersets. For example, any strong limit cardinal[0] can't be a powerset. Obviously, no uncountable total order can be isomorphic to a subset of the natural numbers. You can then well-order that strong limit cardinal to get a total order which isn't a powerset and isn't isomorphic to a subset of the natural numbers.
Second, even countable total orders are much more varied than subsets of the natural numbers. For example, the integers form a total order which can't be isomorphic to a subset of the naturals. The integers are unbounded below, but any subset of the naturals is bounded below (by 0, e.g.).
As another counterexample, the set of rational numbers in [0, 1] forms a total order which is dense: between any distinct elements of the total order, there's another distinct element between them.
You can get a nice theorem along these lines, though. Every countable total order is isomorphic to a subset of the rational numbers. [1]
Of course, the word "most" here is ambiguous, but seeing as there are uncountably many non-isomorphic, countable, total orders (for example, the number of countable ordinals is uncountable), but only countably many non-isomorphic subsets of the naturals, I think it's inappropriate.
[0] https://en.wikipedia.org/wiki/Limit_cardinal [1] https://www.whitman.edu/mathematics/higher_math_online/secti...