There is no intrinsic need for ultrafast pulses $-$ it is perfectly reasonable to think about implementing this in other spatial domains, from quasi-CW lasers in the visible domain all the way down to radio frequencies. (We're hoping that the OAM community will try this, of course, and that they'll help us figure out cool ways to use this in other spectral domains.)
[Edit: they did! Beams with self-torque have now been re-implemented in a separate spectral domain (at the telecomms wavelength, 1.5 $\mu\rm m$, with self-torque timescales of the order of $1\:\rm ns$, some five orders of magnitude slower than our original work), as reported in Nanophotonics 9, 2957 (2020).]
The simple reason for why we used ultrafast pulses is that the concept developed naturally out of work in this area. Specifically, the configuration that produces self-torqued high-order harmonic emission is a natural extension of the previous work by Laura Rego and the Salamanca team [Phys. Rev. Lett 117, 163202 (2016)] (itself a natural extension of previous OAM work in HHG), where they looked at the HHG produced by a superposition of $\ell=1$ and $\ell=2$ beams, and showed cool nonlinear effects. To go from there to the self-torque configuration, you just need to add a time delay between the two, which is one of the standard tweaks in ultrafast science.
(That said, once you add in that tweak, you still have to figure out what it is that you've made, and what it means, and how you could possibly measure it. And the experimental campaign to measure it was no walk in the park, either. These were major breakthroughs by the Salamanca and JILA teams, respectively $-$ I helped analyze the results of the simulations.)
Still, I would argue that there is a deeper reason why this new perspective came up within HHG and not elsewhere: nonlinear optics, and particularly extreme nonlinear optics, is different. It's bigger. It's harder. It forces you to think in new ways. And it's definitely cooler, of course!
More to the point, the habit of thinking about optics exclusively in terms of monochromatic light very often makes you lose sight, completely, of the time-domain aspect of the system; this gets you some powerful tools, but it also filters out useful information. HHG does not afford us this luxury: we have to think constantly in terms of the time-domain behaviour of light, and this brings to the surface a number of interesting features that are just completely lost by working with monochromatic light or in the frequency domain. Optical self-torque is an excellent example of this.
(And, since I'm in shameless-plug mode: this other recent paper (arXiv) is a similar example, where the time-domain perspective on multi-colour combinations, as developed in HHG for the 'bicircular' fields used to produce circularly-polarized harmonics, helped us uncover a completely new optical singularity with cool new symmetries $-$ and, in the process, to break the results that sparked this previous question. And this, again, plays nicely with HHG and its conservation laws (arXiv).)