There are four questions here (including the one in the heading) that appear to involve aspects of the history of calculus methods. For clarity in case of future amendment, the questions as they are addressed here are:
(Heading) Did Berkeley's criticism of infinitesimals hobble calculus pedagogy?
(This heading-question was also amplified/explained in the text: [Did] Berkeley's criticism [lead] to ... revisions to calculus textbooks, including ... more rigor -- and [did] those changes and additions [make] calculus harder to teach, learn, and understand.)
(1) Is the authors' premise that calculus would be easier if we dumped the formalism correct/reasonable?
(2) If we were to drop the formalism, would there be any downsides for non-math majors?
(3) Was it really Berkeley's critique that caused this shift, or would it have happened anyway?
First, about questions (Heading) and (3):-
Berkeley's (1685–1753) criticism (in "The Analyst; or a Discourse Addressed to an Infidel Mathematician" (1734)), which targeted Leibnitz's infinitesimal calculus in L'Hopital's version of 1696, was the third of three widely-known critiques of calculus-like methods that appeared from about the mid-17th to the mid-18th century. All three critiques had a similar central point: to condemn the ambiguity or contradiction said to arise from inconsistent treatments of infinitesimals (or earlier, indivisibles). For the critics, infinitesimals seemed to be treated as non-zero for one purpose and then as zero for another purpose in the same demonstration. This was offensive to those who held to an early mathematical ideal, which was to deliver certain knowledge through proofs that were also certain and indubitable. Ambiguities and contradictions were repugnant to this ideal.
On the first occasion of such critiques, in the mid-17th century, the target was the 'method of indivisibles' proposed by Bonaventura Cavalieri (c.1598–1647) in 'Geometria indivisibilibus continuorum ...' (1635), and then, with a reponse to criticisms, in 'Exercitationes geometricae sex' (1647). Critics here were Paul Guldin (1577-1643), in vol.4 of his "Centrobaryca seu de centro gravitatis trium specierum quantitatis continuae" (1635-1641), and André Tacquet (1612-1660). Cavalieri's method remained under a cloud of doubt and disrepute even though it was acknowledged (e.g. by Torricelli and Newton) to yield many useful results.
On the second occasion, at the end of the 17th century and at the turn of the 18th, the main critic was Michel Rolle (1652-1719), who vigorously attacked, especially, the Leibnizian infinitesimal methods published by L'Hopital in 1696 in 'Analyse des infiniment petits'. His attacks have already been discussed in this forum, in Did Michel Rolle say that the calculus is "a collection of ingenious fallacies"?. On that second occasion, the attacks were conducted in public in Paris and were defended among others by Pierre Varignon, whose arguments were notable for their appeal to Newton's method of limits (the 'method of first and last ratios' given in Book 1 section 1 of the 'Principia', see below.)
Berkeley's 1734 version of the critique did have effects on the writing of at least one textbook: 'Treatise of Fluxions' (1742) by Colin Maclaurin (1698-1746). This book was regarded by Jean-Étienne Montucla (1725-1799) in his Histoire des Mathématiques (2nd ed., vol.3), as an effective rebuttal of Berkeley's criticism and as a good defence of the calculus methods of both Newton and Leibnitz.
Montucla's 'Histoire' gave this special appreciation of Maclaurin's defence of the calculus (in passages from pp.116 and 118, here in my translation):--
"Nobody is unaware these days that the infinitesimal calculus is in
its fundamentals absolutely the same as what Newton has called the
calculus of fluxions. Now, this latter has nothing that is not in
conformity with the most rigorous principles of Geometry, as has been
shown at full length. Accordingly, both the one and the other must
enjoy the same degree of certainty."
[...] "It was seemingly in answer to the attacks of Berkeley that Mr
Maclaurin undertook his 'Treatise of Fluxions' that appeared in 1742.
There, the method of Newton is fully demonstrated without any
assumption of infinitesimals or anything else capable of lending
itself to controversy ... Mr Maclaurin's demonstrations are of
prodigious length ... he could have limited himself to some examples
... . [But] however that may be, one can say that if any doubts could
remain about the solidity of Newton's method, they are entirely
dissipated by this work of Maclaurin ... ."
So one answer to questions (Header) and (3) can be this: that Berkeley's criticism, while of immediate influence, may not have been determinative of the course of calculus criticism and exposition, since other criticisms of similar kind had already been published, and were 'in the air' and known widely, even if not well, before him. On the other hand, Maclaurin's response to Berkeley was very lengthy and not easy to understand.
In response to question (2):
The controversies about calculus-related methods have mainly concerned their justificatory foundations, with much less impact on the careful practice even of some methods impugned by the critiques.
For example, in relation to the method of indivisibles, Cavalieri's successor Evangelista Torricelli (1608-1647), "worked with indivisibles, but he cautioned that their uncritical use could lead to paradoxes" (see "A History of Mathematics" (3rd Edition, 2008) V J Katz). Thus, it was probably in cautious recognition of shaky foundations for the method of indivisibles, that Torricelli took precaution after obtaining by their use -- and doubting -- a surprising result, namely, that an infinitely long solid of revolution could nevertheless have a finite volume. Thus, he found through indivisibles that the infinitely-long solid formed by rotating a hyperbola xy = k^2 around the y axis from y = a to y =∞ nevertheless has finite volume. Torricelli then quieted doubts and confirmed the result by presenting a second derivation, relying on the indubitable classical method of exhaustion.
Likewise, Newton in his 'Principia' declined to rely on the dubious justification of the method of indivisibles for his calculus methods, even though he stated specifically that 'hereby the same thing is perform'd as by the method of indivisibles' (Newton 'Principia', 1729 translation, vol.1, Book 1, sec.1). He gave reasons for setting out his method of limits ('first and last ratios') :-
"These lemmas are premised, to avoid the tediousness of deducing perplexed demonstrations ad absurdum, according to the method of the ancient geometers. For demonstrations are more contracted by the method of indivisibles : But because the hypothesis of indivisibles ... is reckoned less geometrical; I chose rather to reduce the demonstrations of the following propositions to the first and last sums and ratio's of nascent and evanescent quantities, that is, to the limits of those sums and ratio's; and so to premise, as short as I could, the demonstrations of those limits. For hereby the same thing is perform'd as by the method of indivisibles; and now those principles being demonstrated, we may use them with more safety."
So a possible answer to question (2) would seem to be this, that anything that only expresses later work on justificatory foundations for calculus-like methods might be disregarded in practical use (leaving the user in good historical company) -- provided that any shortcuts are applied with care inside the limits of well-established practice for their use.
(In view of the above, this answer does not detail 19th-century work on foundations of calculus, especially by Cauchy and Weierstrass: it seems enough for present purposes to limit the discussion to the 17th- and 18th-c. history which shows the main lines on which controversies developed. Again, the 20th-century non-standard analysis appears to concern justificatory foundations rather than everyday practice.)
(As for question (1), the questioner could not identify the author or the details of the author's suggestion: nothing can be added about that here.)