I know exactly what being a constructivist means - I work with them every day.
The only people who take issue with descriptions of infinity are contrarians and laymen, who we ignore and therefore do not even consider, and constructivists, who have relevant points to make but are too busy getting hung up on semantics to make their points, and often fall back on classical argument anyway.
is a perfect example. At the same time, Spivak tells you that "infinitely small changes" between coordinates are nonsense, and then immediately tells you, "But you can (almost) describe them with a tangent vector". Are they nonsense, or is our description nonsense? He's not remotely arguing that infinitesimals don't exist or are nonsensical - only that our description of them is inaccurate. Spivak, being the actual genius he is, makes a very, very strong argument for why representing trying to represent infinitesimals directly using classical annotation is inaccurate.
Unfortunately, the same way most mathematicians are actually just idiots pretending they are smart because other very intelligent people have worked in mathematics, most constructivists don't understand the difference between a critique of a representation and a critique of the thing itself, and will argue to you that you must be aware of this difference while conflating the two, simultaneously.