How can a crossed dipole (turnstile) antenna be omnidirectional in 3D?
The author of that paper themselves didn't even fully read the survey paper they're citing: The antenna they replicated was a circularly polarized one, not an omnidirectional one.
When you think about symmetry and what circular polarization means, you'll quickly realize you can't have omnidirectional emission and consistent circular polarization on a sphere; you need the polarization to change, which means that the field can't have the same magnitude in all polarization planes all over the sphere.
In fact, that's a corollary to the pretty intuitive &quot;Hairy Ball Theorem&quot; by Poincaré (the same person of Poincaré Sphere fame, the thing you've learned when studying mixed polarizations).
So, this literally can't be an omnidirectional antenna.
Hope this is authorative enough. For general commentary on how to approach papers that claim great things about antennas, but don't offer proof, see my other answer.

By the way, here's the figure from the paper 53 that they say they can't access:

More than 30 dB attenuation at 135° doesn't look like an omni antenna, does it?
The authors themselves don't claim this antenna to be three dimensional omni. Quite the contrary

A three-dimensional (3-D) plot of the far-field
resembles a doughnut with an omnidirectional distribution in
the plane perpendicular to the dipole axis and a figure eight
pattern in the orthogonal cut.


TL;DR: doesn't work.