I don't know if all horns have similar patterns or if you need one of those orange wafers (phase plug) inside.
Some horns have similar patterns, but the radial phase plug has little if anything to do with the exceptionally uniform radiation pattern of the Altec Model 14 as shown in their ad. Rather, the controlling factor is the geometery of the horn that Altec used, which they called the "Manta Ray".
The Manta Ray was a constant-directivity diffraction horn; that is, the horn had deliberate, aggressive discontinuities
(internal slots and/or sharp angles or edges) which strongly diffracted the output causing the pattern to widen. So instead of the sound "beaming" at high frequencies and being concentrated on-axis, it was spread around fairly evenly within the coverage pattern. Since then many companies have used diffraction horns.
Let me back up and give a brief and incomplete overview of horn designs.
In the early days when amplifiers were weak, the most important characteristic of a horn was acoustic amplification. These early horns were typically very long and large, and had narrow radiation patterns. Their length and large mouth size helped them to be effective down to lower frequencies.
Sectoral horns were an early attempt to widen the coverage pattern. These use slots or vanes or cells to help spread out the high frequencies more uniformly.
Diffraction horns like the Manta Ray use diffraction to spread the highs around, and diffraction horns have been widely used in prosound ever since, and are used in some high-end audio designs. Altec was one of the first to also incorporate "pattern matching" in the Model 19/Model 14 series. This is where the woofer's radiation pattern approximately matched the horn's pattern in the crossover region. The Model 19 used the same woofer and horn as its predecessor, the Valencia, but the crossover frequency was considerably higher, so that the woofer's pattern had narrowed enough to match the horn's pattern. The landmark JBL model 4430 studio monitor came along a few years later and was arguably a more "purist" application of this concept.
Meanwhile over in the high-end home audio world, much more gentle horn profiles began to show up: Tractrix, Exponential, Hyperbolic, Spherical, Le'Cleach, and so forth. These all have a trumpet-like profile, with a very gentle curvature back near the compression driver, and then the curvature accelerates near the mouth. One characteristic of this sort of profile is that the directivity of the high frequencies is controlled by the walls of the horn in the first few inches (where the angle is narrow), while the directivity of the lower frequencies is controlled by the walls of the horn near the mouth (where the angle is wider). So typically their radiation pattern gently narrows as we go up in frequency. One advantage of concentrating the highs into a smaller angle is higher efficiency; the amount of high-frequency energy available at the top end is often a limiting factor as far as the efficiency of a horn system goes, and a narrow pattern in the top octave results in higher on-axis efficency than you would get with a diffraction or multicell horn.
Waveguide-style horns are focused on radiation pattern control rather than acoustic amplification. Their profiles have most of their curvature near the throat, except for the round-over at the mouth. They tend to have pattern uniformity rivalling diffraction horns, but don't offer as many pattern shape options as diffraction horns do. Like diffraction horns their highs are spread out over a wider area, which lowers their on-axis efficiency at the top end. Waveguide-style horns tend to be fairly short so they don't have good acoustic loading down very low, but usually they are pattern-matched with a woofer anyway so it's seldom an issue.
I have tried to avoid comments about horn sound quality here; that's a complicated and controversial topic. Perhaps another day.
I'm not sure what the correct search terms would be or if one of our experts has a few images they could post, but I'm not seeing in my searches what I want regarding: polar plot, dispersion radiation pattern, lobe, directivity, polar response and so forth.
Imo the most informative type of directivity graph is the polar map, which uses color to convey intensity. Here is a paper on the subject:
http://www.gedlee.com/Papers/directivity.pdf
Audioholics uses polar maps in some of their reviews; here's one:
https://www.audioholics.com/bookshelf-speaker-reviews/monolith-k-bs-1/measurements
I want a side by side comparison of all the major speaker types.
There is so much variation from one model to another that only the broadest of generalities could be conveyed, with a veritable plethora of exceptions being the norm. And it is often these exceptions which are the most interesting.
Still I can see how what you are asking for would be useful. If I come across something along those lines I'll post a link.