Sorry, I should have actually bothered to read the section before commenting. But very nice that the expression for $<t>$ is exact!

"...we're neglected the latter summation/integral term as a small positive constant that does not scale with the diffusion coefficient..." This doesn't matter, but is it safe to assume that this term is $<<1$?

Ok, I see. Regarding the unnumbered equation after Eq. 57 in arXiv:1101.5043: we're assuming the unit radius sphere, otherwise it's: $\langle t\rangle=\frac{2R^2}{D_{r}}...$, we're dropping the terms involving contributions from $D_2$ (related to bulk diffusion) (which seems OK to me), and we're neglected the latter summation/integral term as a small positive constant that does not scale with the diffusion coefficient.

This answers my question! However, if you could point out where $<t>$ was derived (unless you did it yourself) that would be very helpful for me in terms of understanding.

Also, is the expression for $<t>$ starting from $\theta_0 \in (0, \pi)$ from the linked arXiv paper? I can't seem to find the derivation for it in the 3D case section.

I agree with your interpretation of the problem. However, I'm a little confused about your expression for $<t>$ provided random uniform initiation, shouldn't there for a $D_r$ term somewhere?