(Edited after comments by mods. Thanks for guiding me through the process!)
The only thing that is elliptical is the planform shape of an elliptical wing.
Many people unfortunately confuse an elliptical pressure distribution with the
pressure distribution over an elliptical wing. They are not equivalent.
On an untwisted elliptical wing the local lift coefficient is not constant over
span, as some here and in many other places on the internet maintain.
Arguments from lifting line theory are not appropriate for discussing the
behaviour of the flow near wing tips. It is unreasonable to expect lifting line
theory to be valid near the wing tips because is not a consistent large aspect
ratio asymptotic expansion.
See: Van Dyke, "Perturbation methods in fluid mechanics", 1964.
The span loading includes a logarithmic term, hence it is not elliptical.
Furthermore, it cannot induce a constant downwash, and so at the trailing edge
the vortex wake does not start as a flat sheet.
The process whereby the vortex sheet rolls up is far more vigorous than if it
started out as a flat sheet because the relatively weak effect of viscosity
would be the main mechanism driving that rolling-up process. There is a strong
upwash in the flow-field near to and around the wing tips which initiates the
rolling-up process far more vigorously.
See, for example, among many other papers:
Peter F. Jordan, "Exact Solutions for Lifting Surfaces", AIAA Journal, Vol. 11, No. 8, 1973., pp. 1123-1129.
Peter F. Jordan, "On Lifting Wings with Parabolic Tips", ZAMM 54, pp. 463-477, 1974.