I would say "horn-loaded" as the driver feeding the horn could be any of several different types, including conventional cones. Separately, a driver can be back horn-loaded, as in the Tannoy Westminster Royal GR.
I am much more inclined than most to use higher than typical power on high sensitivity loudspeakers. So I am answering your questions more from the point of view of horn loudspeakers than from the point of view of SET amplifiers.
I am answering these questions having in mind the very best and most ambitious speakers of each topology I have heard.
1. In which areas to SET/horn systems excel?
midrange realism, naturalness and presence
reproduction of the sounds of brass instruments
dynamics and "jump factor"
wider listening zone (no narrow "sweet spot")
Horn-loaded bass can sound very "fast."
An overall sonic presentation which is different from any other speaker topology in that horn speakers project the sound towards you in an energetic way that contributes to "aliveness," presence and realism.
2. In which areas can they not even be matched by other types of systems?
the reproduction of the sounds of brass instruments
I think there is something about the way horn-loaded loudspeakers reproduce the sounds of brass instruments which is consonant with the way those instruments themselves produce sounds.
If my primary musical genre interest were jazz, I definitely would have a horn speaker system. If my primary musical genre interests were jazz and classical, I probably would have a horn speaker system (unless I could have a Wilson XVX + Subsonics in which case it would be very difficult to make a decision).
3. Are there any sonic drawbacks to SET/horn systems, in the sense that there are other system types that just can do certain things better?
I find almost all contemporary design horn speakers to be a bit aggressive-sounding and fatiguing. The Tune Audio Avaton and the Cessaro Zeta and the Pnoe are my only three exceptions to this personal subjective experience. For example, I have yet to hear any Avantgarde or Acapella speaker system which I did not find a bit fatiguing.
I continue to feel that, for me, there is something slightly more transparent and in-the-room real about the way planar speakers reproduce vocals than any other speaker topology I have heard.
I think big cone speakers systems with a lot of driver surface area in the midrange and from the midrange on down can reproduce piano and big classical symphony orchestra at least as well as big horn speaker systems.
4. How much money do you have to spend on an excellent SET/horn system? How much money do you have to spend on a good SET/horn system?
Small horn speakers don't do the trick for me. I feel the same way about planars.
Unfortunately, with contemporary designs the horn loudspeaker system has to be big (and this inevitably means expensive) to achieve the sonic attributes I am attributing to them. (I have never heard from a 30" tall, two-way JBL or similar what I have heard from big horn systems.)
5. What about the question of the oft-mentioned horn colorations? Are they still a concern in modern horns? Are they a concern in vintage horns?
I don't know what you mean here by "colorations." (Is ruler-flat frequency response which sounds unnatural and unrealistic a "coloration"?)
I think some people refer to horn "colorations" as the horn "shouty-ness" phenomenon or the "cupped hands" phenomenon. I don't hear either of these on the big horn loaded speakers I have heard.
I will define "coloration" here as frequency anomalies. Other than the Shindo system Keith took me to hear in Hollywood, which sounded to me like bourbon dripping down a warm brownie, I do not think horn-loaded speakers have any ubiquitous infirmity regarding colorations.