Suitability of standard-gain pyramidal horn for UWB signal
Now that's a good, hard question, and well phrased! Disclaimer: my antenna engineering experience is 40 years old, but if physics hasn't changed, maybe I can help you some.
First, you're right, for this application, avoid log-periodic antennas in favor of antenna designs with a single, broadband feed point, such as the horn you describe. You will have a constant &quot;phase center&quot; over the horn's operating bandwidth.
But you can think of the horn as a special kind of waveguide whose dimension varies along its length. And the horn, like any waveguide, will exhibit &quot;frequency dispersion&quot; -- that is, waves propagate through the horn at a velocity that varies with frequency, slowing down especially as you get close to the horn/waveguide's lower cutoff frequency. (I'm skipping over the fine points of group velocity vs phase velocity here; you care about the UWB modulation on the wave, which is affected by group velocity.)
Practical thoughts:

If you don't need directivity, consider a broadband dipole instead of a horn.
If you do need some directivity, choose the shortest, least directive horn available, as it will exhibit less dispersion.
Operate as far from the lower-frequency cutoff of the horn as possible
If possible, find a horn with an exponential taper, rather than a linear taper.