What did the logicians of the 20th century think? Perhaps this was best described by Michael Beeson in his book ("Foundations of constructive mathematics: metamathematical studies", 1985, Springer):

The thrust of Bishop's work was that both Hilbert and Brouwer had been wrong about an important point on which they had agreed. Namely both of them thought that if one took constructive mathematics seriously, it would be necessary to "give up" the most important parts of modern mathematics (such as, for example, measure theory or complex analysis). Bishop showed that this was simply false, and in addition that it is not necessary to introduce unusual assumptions that appear contradictory to the uninitiated. The perceived conflict between power and security was illusory! One only had to proceed with a certain grace, instead of with Hilbert's "boxer's fists".

So, apparently they were both too pessimistic. Of course, they were not so naive as to think that bridges would fall down if a switch to constructive foundations were made. They just expected the constrictive foundations to be insufficient (in the same sense as, say, ancient Greeks' notion of number was insufficient for carrying out real analysis).

Let me also say that it is not a trivial matter to establish what sort of foundations were used by Newton, Leibniz, Weierstraß, Cauchy, etc., because they did not present their work in the way we usually do today. It is perhaps most honest to just admit that their notion of foundation was not of the same kind as ours. Nevertheless, at least all the calculational proofs (and that would be a large proportion of all proofs) would be naturally constructive already. Fundamentally non-constructive reasoning came later, with Hilbert's proof of Nullstellensatz and the non-constructive principles of set theory (such as the axiom of choice).

And one should not commit the logical error of thinking that just because pre-20th century mathematics could be retrofitted with classical first-order logic and Zermelo-Fraenkel set theory, it must be so retrofitted.

What did the logicians of the 20th century think? Perhaps this was best described by Michael Beeson in his book ("Foundations of constructive mathematics: metamathematical studies", 1985, Springer):

The thrust of Bishop's work was that both Hilbert and Brouwer had been wrong about an important point on which they had agreed. Namely both of them thought that if one took constructive mathematics seriously, it would be necessary to "give up" the most important parts of modern mathematics (such as, for example, measure theory or complex analysis). Bishop showed that this was simply false, and in addition that it is not necessary to introduce unusual assumptions that appear contradictory to the uninitiated. The perceived conflict between power and security was illusory! One only had to proceed with a certain grace, instead of with Hilbert's "boxer's fists".

So, apparently they were both too pessimistic. Of course, they were not so naive as to think that bridges would fall down if a switch to constructive foundations were made. They just expected the constructive foundations to be insufficient (in the same sense as, say, ancient Greeks' notion of number was insufficient for carrying out real analysis).

Let me also say that it is not a trivial matter to establish what sort of foundations were used by Newton, Leibniz, Weierstraß, Cauchy, etc., because they did not present their work in the way we usually do today. It is perhaps most honest to just admit that their notion of foundation was not of the same kind as ours. Nevertheless, at least all the calculational proofs (and that would be a large proportion of all proofs) would be naturally constructive already. Fundamentally non-constructive reasoning came later, with Hilbert's proof of Nullstellensatz and the non-constructive principles of set theory (such as the axiom of choice).

And one should not commit the logical error of thinking that just because pre-20th century mathematics could be retrofitted with classical first-order logic and Zermelo-Fraenkel set theory, it must be so retrofitted.