First some history and then an opinion
I remember buying my first electronic calculator in the late 1960s. It did addition, subtraction, multiplication, and division, and could remember one number. It displayed 8 digits and had a special button that displayed the next eight digits. I remember using those next eight digits never. To buy it I had to drive 40 minutes and spend $70 – expensive even today.
The square root of 743 computed using the algorithm discussed in the post. The third iteration (fourth answer) is correct to 10 digits.
With it I learned an iterative algorithm for finding square roots: guess the root, divide the guess into the number, average the quotient and the guess, repeat using the average as the new guess. You could do it all without writing anything down. (See the illustration on a modern calculator – accurate to 8 decimals in only 3 iterations (fourth answer), but then I could find the next 8 with the special button.)
Since then, I’ve had lots of calculators of all sorts.
Graphing calculators hit the general market around 1989 or 1990. This was the same time as the “reform calculus” movement. The College Board announced that the AP calculus exams would require graphing calculators in 1995 – five years to get the country ready.
The College Board held intensive training immediately following the reading. These were the TICAP conferences (Technology Intensive Calculus for Advanced Placement). Half the readers were invited for the first year and the other half for the second, then more for the third year.
Casio, Hewlett-Packard, Texas Instruments all gave participants calculators to use take home. Sharpe lent them calculators (and we haven’t heard of Sharpe since). Sample lessons were taught using Hewlett-Packard CAS calculators and then the same lesson was taught using TI-81s. The HP computer algebra system calculators, with far more features but using the far more complicated reverse Polish notation entry system, lost in the completion to the simpler to use, but less sophisticated TI-81s.
The teachers were not all happy. A friend of mine, due to retire in 2-3 years gave up his AP calculus classes early so he would not need to learn the calculators. Others embraced technology. The AP program forced the graphing calculator into high schools where they were used to improve learning and instruction. Yet even today not all high schools have embraced technology.
The calculator makers, especially Texas Instruments, provided print materials, software, workshops and conferences that helped teachers learn how to use graphing calculators in their classes at all levels.
Technology, as a way to teach, learn, and most importantly, do mathematics, caught on big time. And that was and is a good thing.
I think graphing calculators are very quickly becoming obsolete and should be phased out.
Technology has bypassed graphing calculators. Tablet computers, PCs, Macs, iPads, and the like, even smart phones, can do everything graphing calculators can do. They are more versatile. The larger screens are easier to see and can show more information without crowding.
The initial investment may be more than for a graphing calculator, but once purchased the apps are relatively cheap. There are many free apps that not only do computations and graphing, but CAS operations as well. Interactive geometry and statistics apps are also available.
These, along with online textbooks and internet access, put everything students need to learn math literally at their fingertips. Graphs and other results can be easily copied and printed, or pasted into note-taking apps.
One disadvantage is the initial cost for the hardware (but of course many students already have the hardware). The other disadvantage is the ability to communicate and find help both in the room and around the world during tests. Photographing the questions for later use by others is another concern. I think (hope) it is just a matter of time before this problem can be overcome perhaps with an app that allows access only to the apps the teachers allow for tests.
Technology, like time, marches on.
Teaching Calculus 
I really don’t see the rationale for having them. It feels like some naive “educator” fad to have some of that them thar Technology. But more about the image of things than the substance. There is a reason why calculators haven’t caught on or been pushed as much in universities. You don’t really need calculators to get the key concepts of calculus down. And once, you do, if you need to run a computer program in an engineering class, you’re better off doing that. And it’s not like the “TI time” is going to make you better at doing programming algorithms.
I would totally contrast this with chemistry. Here calculators make all kinds of sense as you are routinely crunching awkward numbers in stoichiometry, gas law, equilibrium, etc. problems. This is an are where scientific calculators made a rapid advance. With very little controversy. And similarly at both high school and college levels. But calculators in calculus? Totally silly. Concentrate on learning the different tricks (e.g. trig subs) and on being able to do multi-step algebraic problems (e.g. partial fractions).
The whole thing feels like neoliberal ETS faddishness. They are losing sight of their key mission, which is to replicate a college class. NOT to try to reinvent it because they think they know better.
Oh…and don’t get me started on Common Core. The idea that we need “standards” is silliness. The issue is not the “what”, the issue is the “how”. Pedagogy is more important than curriculum. But neoliberals think they are smarter than they are. McKinsey, Chelsea Clinton. Ugh.
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One of the reasons graphing calculators and CAS calculators were adopted by the AP Calculus program is to lessen the “[concentration] on learning different tricks [your word] and on being able to do multi-step algebraic problems (e.g. partial fractions) [which are still required in AP Calculus]”. The push came from college professors who found their students could to any differentiation or integration they were given but had no idea what a derivative or definite integral meant or was used for.
The AP Calculus program, with calculators, is aligned with most college programs. The College Board and ETS every few years looks at and revises the course description to keep it up to date with what their clients (colleges and universities) are doing. The most recent revision went into effect in 2019.
As you mention, once you know the concepts you can run a computer program (i.e. use technology) to do the grunt work. Learning how to do complicated antiderivatives is not learning how to use a definite integral to solve a real problem.
You may be interested in reading Algematic, a piece I published almost 20 years ago that is, I think, still relevant.
My own problem with graphing calculators is that they are so 20th century. Apps available on tablets and computers themselves can do the same work much better and easier than handheld graphing calculators.
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