The OP’s statement on the matter is this:
Seems like it is more likely that the relevant difference for the student is between computational and applied courses (typical for lower division math courses shared with other majors) versus proof and theory oriented courses (typical for upper division math courses that mostly math majors take), rather than the calculus / analysis versus algebra subareas.
I have a different take on her descriptions. The two types of math are very different. People who fall in love with abstract algebra often find analyses (including complex analysis) messy, tedious, and inelegant (relatively, of course), and proceed along a different path. OP will find out soon if her son likes complex analysis as much he liked abstract algebra.
The article below written by a Hamilton student describes a highly collaborative math department, much like the ideal wished for in the opening post.
I can’t personally evaluate the extent of Hamilton’s math offerings, but the article notes Hamilton’s high concentration of math majors and the “enormously gifted students” who have graduated from the program.
To me it seems that OP’s son is enjoying his ability to visualize multi-dimensional functions. How you solve the problem (calculus vs algebra) is less important than the (rare) ability to visualize the problem itself. Personally I preferred proofs by induction, but I wasn’t very good at either calculus or algebra 
I’m always fascinated by what a “highly collaborative” maths program involves. We were strongly discouraged from any collaboration in solving problem sheets, and it was regarded as cheating if you asked someone else for help. It was pointless anyway since your entire grade depended on the final exam.
If collaboration potentially enhances the learning process, then it is encouraged at Hamilton. Exams are taken independently. When this approach has been tested externally, as in the international Putnam exam, Hamilton teams have, at times, performed notably well in competition with those from much larger schools.
What do you mean by notably well? They haven’t placed in the top-5, ever, the top-5 year in and year out are MIT, Harvard, Stanford,followed by UCLA, Princeton, Columbia, Berkeley, Princeton.
29th in the international competition, which includes schools that you noted with undergraduate enrollments over 15 times that of Hamilton.
How is this different from upper (or graduate) level pure math in any other math department? It is not like these are pre-med courses.
The environment appears to differ substantially from that which @Twoin18 described above, regarding whom my post was a response.
Not sure why you are pushing Hamilton, since it has only 9 upper level courses ordinarily called “pure math” (304, 314, 315, 318, 322, 324, 325, 361, 363), 2 of which the OP’s student has already taken the equivalents of at UCB (318, 325). The student will probably take a few more at UCB before graduating high school, so if the student attends Hamilton, he will probably run out of courses to take.
https://www.hamilton.edu/academics/departments/Courses-and-Requirements?dept=Mathematics
Collaboration doesn’t mean students copying each other’s work. If students are in an environment more or less with their true peers, collaboration could mean a peer group could be assigned more challenging problems to work together on for their mutual benefits. As long as each student’s performance can be measured individually outside of group efforts, I don’t see a problem with collaboration.
What are you talking about? It appears as if you haven’t read the thread. Any comments I’ve made on this topic have been based on the course descriptions of several LACs I have researched. From this group, I have not recommended any in particular, nor have I even recommended LACs as a class for this OP.
You seem to be pushing Hamilton in this post #126 (and #128):
I explicitly recommended up-topic that the OP should contact the department chairpersons at smaller colleges of potential interest. What you quoted above was, as stated once already, simply a response to another poster on a side-topic that appeared to be of genuine interest to them.
With respect to replies directed toward the OP, my most recent suggestion was to include graduate department rankings in her research through the link provided.
I did not read the whole thread OP, but my understanding is that your kid is of the “self-taught mathematician” types. Any college/university with a good library, access to math papers and a professor or two that would be willing to advise would probably fit the bill. Since your kid is already reading math papers, they should look into the authors’ affiliations and that should give them an idea of where some good math research is taking place. It’s a different story if your kid needs to sit in math instruction to move along.
“29th in the international competition, which includes schools that you noted with undergraduate enrollments over 15 times that of Hamilton.”
You’re playing a little loose with the facts in your effort to sell Hamilton. Hamilton has about 60 math majors, MIT does have more, around 200, but not 15x more. Cal Tech has around 60., Stanford around 120.
“As long as each student’s performance can be measured individually outside of group efforts, I don’t see a problem with collaboration.”
That’s easier said than done though, you may be able to collaborate on problem sets, but it wouldn’t be forced. You could I guess divide the class into study groups, but there’s no way to ensure each group is equal or similar wrt math abilities. And if a student can solve it by themselves, and many can, you really should make it optional.
Here’s what I had in mind:
A portion (say 20%) of the course grade is to be based on collaborative effort. Some of the most challenging problems could be desinated to be worked on by students collaboratively. There’re almost always multiple ways to solve a problem, especially a math problem. By discussing their solutions, even students who could have solved the problem on their own might learn a thing or two. Some students would surely contibute more and some less, but the idea is to encourage collaboration. Exchanging ideas and collaborations among students are part of learning. They should be encouraged in a similar way that proferssional researchers are encouraged to exchange ideas at conferences and collaborate on projects.
@theloniusmonk: You asked a question seemingly without a sincere interest in a reply. UCLA and UCB (schools you offered as exemplars) enroll easily over 15 times the undergraduates of some liberal arts colleges. The Putnam Competition does not promote itself as being exclusive to math majors, so students with majors in other fields (e.g., engineering) may participate (and by this consideration of relevant majors, Caltech would enroll over three times as many students as an LAC). I did like the article that another poster linked on this thread, in that it does depict — for the OP, as well as for those reading through for general information — a collaborative math department at its best. However, although I shouldn’t have to say it again, my most recent suggestion to the OP on this topic was to research schools with highly regarded graduate departments. Going back, my first post on this topic was subjectively arranged, from an alphabetical original source, to list schools with graduate department opportunities, which might be more suitable for the OP’s advanced level, closer to the front (with some additional ordering by selectivity, geography, etc.). If I were, as you and another poster (as well as a “like” type) seem to believe, favoring one school (or class of schools) over others, subjective aspects in my posts such as this would have differed.
Obviously, I have not been very encouraging on this thread about the appropriateness of LACs for advanced math students. Given inevitable scheduling conflicts, and the frequency with which what advanced courses are available are actually offered, it’s just a tough slog to fill the four years with appropriate and interesting courses, especially for the student who is more interested in the “purer” topics.
That being said, our S21 has applied to Amherst regular decision, the school is upfront about possible pathways for the advanced student (ours would come in just above the intro analysis level, but with some substantial work in number theory and combinatorics). Look at the pathway here under “Advanced Students” from the Amherst catalog and you will see how it relies on nearby UMass in the consortium:
OP’s son would likely be a little more advanced than our own upon entry, and is even more interested in taking more math courses than required for the major. Open curriculum Amherst is worth a look, but again any T20 university I think would be better, as would a number of state publics.