As a matter of fact, there are contexts where writing numbers in octal is not just "beautiful", but useful, and is actually done. See asciitable.com and the Unix chmod utility (chmod 755 script.sh) for some common examples off the top of my head. And, of course, programmers and computer scientists deal with numbers in binary and hexadecimal all the time, and you do find numbers written in such bases in academic and other papers.

Also, the question "How many factors of the prime p are in n factorial?" is easy to answer if n is written in base p (0 * [ones digit] + 1 * [p digit] + (p+1) * [p^2 digit] + (p^2+p+1) * [p^3 digit] + ...), and one can take this in the other direction--"Find the smallest n for which p^k divides n"--without much difficulty. This is how I'm planning to answer a particular Project Euler problem, and if I were writing up my solution, I would write some numbers in base p.

Methinks you picked a really bad example to illustrate your point.

The things I find harder to do in bases other than 10 are: recognize numbers (primes/squares/cubes/triangular/Fibonacci/pentagonal/etc), factor them, do arithmetic (though I'm fairly accustomed to doing it in binary), and judge magnitudes. If a paper dealt with these things, and there were no apparent benefits to using octal or whatever but the paper used it anyway, then I'd probably be annoyed. This is probably what you meant to refer to with your example.

Now, you said the numbers would be written in octal "for the sake of beauty". I don't know why someone would find it "beautiful" to write numbers in octal unless they turned out simpler for some reason--e.g. if all the numbers in question turned out to only have the digits 0 and 3. Which would either be a coincidence--and in that case I think a mathematician would find it disappointing rather than beautiful, to see a pattern which turns out not to be robust or to reflect any underlying truth--or it would be the result of some underlying truth, in which case it likely would be better to write it in base 8. The optimal strategy in two-player Nim is best understood and implemented when the numbers in each pile are written in base 2, because you need to compute XORs.

Many of the arguments for tau are based around beauty, yes. This is because many of the arguments for pi are based around beauty. However, there are also usability arguments, which seem to address what you're concerned with. I don't think you can disagree: that it is easier to know of the 3rd and 6th roots of unity as cis(τn/3) and cis(τn/6), instead of cis(2πn/3) and cis(πn/3)--or cis(2πn/6)); easier to reason that a wheel that makes 12 rotations in 5 seconds spins at a rate of 12τ/5 radians per second, rather than 24π/5 radians per second; and easier to remember formulas and facts involving τ... ok, this one seems disputable, but I think I can actually argue for it.

If the only formulas that existed had "2π" printed right on them, then you could accustom yourself to treating "2π" as an atomic concept, and it wouldn't make much difference if we wrote them with "2π" or "τ". However, instead, we have a bunch of formulas with 2π, a fair number of formulas with [some factor]π (e.g. "Coulomb's constant k = 1/(4πε)" and "volume of sphere = 4/3πr^3"), and a couple of formulas with π by itself. This is three patterns to recognize and remember, as opposed to two: τ and [some factor]τ. [1] And your attempts to interpret 2π as a thing in itself will be confounded by your need to interpret π as a thing in itself, both when you think about the formulas with [some factor]π, and when you manipulate expressions and do arithmetic. (We saw above the results of plugging k=6 into 2πn/k; note that this kills the 2π abstraction even though the formula has 2π.)

Basically, your understanding of the circle constant will be fragmented into π and 2π. This was true for me even before I read "The Tau Manifesto", perhaps (not sure) before "π is wrong". If you think in terms of π, you're bound to notice the 2π in the underlying pattern when it comes up; you try to put everything in terms of 2π, and you become disappointed when it fails to simplify some expressions (the 1/(4πε) becomes 1/(2(2π)ε)), and frustrated when arithmetic demands that the 2 be cancelled (obscuring the underlying 2π). You'll probably try for a while; give up, somewhat dissatisfied; and then forget about trying to make sense of the situation. Whereas with τ, you never ever need to think about whether or not this 3π/4 is better represented or thought of as 3/8(2π), whether you should cancel out factors of 2, whether "2π 4π 8π 16π" is really π * 2^n or 2π * 2^(n-1); the cases where the circle constant stands by itself appear without any special arithmetic tricks, and the cases where it's stuck with baggage are immediately plain.

There's probably a reason physicists came up with an entire symbol to represent Planck's constant divided by 2π. It just sucks when your atomic concept isn't an atomic symbol and is likely to get broken up or partially destroyed by arithmetic.

As I've written this, I've become convinced that things really would have felt much better and nicer had I been using τ. So it is important, it makes a significant difference; and this, plus external support (today's event, upvotes, friends' approval, and Vi Hart's video) make me more confident that we will succeed in changing it.

[1] As a side note, what makes the formulas with τ beautiful--or any formulas in general--is the same thing that makes them easier to remember. Fewer different patterns to deal with. Running the set of all formulas in your head through a compression utility would probably produce a smaller output.