I have two C tenors and both of them are just the one key types. I also have two rather rare German tenor recorders built in D, one with no keys and one with a single key.
The recorder appears to persist as an instrument of the really "old school" in some ways. Even though the modern baroque fingering system is not exactly historically correct, the instrument still favors arrangements in C major or modes of C major. Those instruments were built during a time when if you wanted to change keys, you changed instruments. It was that simple. Chromaticity belonged to string intruments without frets or with moveable frets if the concept even existed. Remember;- equal temperament is a very recent concept compared to the entire history of music. And equal temperament is harmonically imperfect;- deliberately so. After only a few hundred years, it should still be regarded as "experimental". It's an experiment that has resulted in a very interesting array of key work and physical and mental calisthenics, patents, patent wars, etc. Once upon a time all wind instruments had some kind of tube with some finger holes. When you ran out of fingers, you ran out of possibilities. Largely, these followed the same kind of fingering rules;- the rules we use when we play recorders. The more keys you remove from any wind instrument, the more like a recorder it becomes. In a sense, it becomes most perfect when it has no keys, or just those required to extend a finger reach. That was the original purpose of keys, to make larger instruments possible that could play a full harmonically accurate scale. Once key work became standard, it was used to create chromaticity in smaller instruments. There is a lot to be learned from recorders.
I am really pleased that someone brought up recorders because since I have been attempting to become an accomplished clarinet player, I have also started playing my recorders more frequently. Conceptually it helps understand the way all woodwind instruments work. Also very useful is my experience with Cherokee flutes, which have a great deal in common with recorders.
There was a time when none of the wind instruments we played were truly chromatic and none were capable of equal temperament. One of the things I want to finally do, is look at the "way" that our early clarinets are "out of tune". If we look at the key the instrument is built for, we might find that optimization for harmonic accuracy in one key persisted well into the late 19th century. It is almost a necessity that this was the case because electronic tuning devices, which I typically discourage the use of, had not replaced the human ear. If a historic instrument plays in tune, it's most likely because the maker could hear in tune harmony. Without a way to calibrate to equal temperament, harmony was probably the guide to tuning the instruments when they were being built. We did not have easily portable electronic tuners until I was a young man. Sets of tuning forks were probably pretty expensive (still are) and even with tuning forks, you have to be able to hear harmony very well. That weird thing in the band room with the flashing dial lights (Conn stroboscope) was not pocket friendly. If you were in the position to need to tune up something like a multi-string harp (or guitar), you tuned to 4ths, 5ths, and had to make a decision about how to handle the major and minor thirds (that pesky B string!).
My working theory, and I really should research this because I am sure others have already looked into it, is that before the 1930s, instrument makers adjusted the tuning of their instruments primarily by ear. What you don't use you lose. Remember when the person that showed up to tune the piano was usually a blind person? That was because blind people develop higher sensitivity to sound. Of course Electronic tuners and click tracks are the death of harmonic nuance and natural rhythm. You may think me a Ludite, but that would put me in company with J.P. Sousa, who felt that the impact of recorded music was such a threat to music being of the people and belonging to most of them, that he went to Congress with his concerns and refused to conduct his band for production of "canned music". You can hear recordings of the Sousa band, but someone other than Sousa is conducting for those.
When someone in an ensemble now says, "Hey we need to stop and tune up", it's the older players that do this. The youngsters don't know if they are out of tune until they check with an electronic device. And they probably are more likely to be on a Pokemon scavenger hunt than checking their tuning with a phone app. The younger generation has almost lost its' ear. Maybe the music was a little too loud? D'ya think? I have a 1972 Marshall 50 Watt half stack. My experience with that is mostly keeping up with my sets of ear plugs. Thank heaven for the quiet recorder and its' persistence as an instrument of both learning and performance. When you need more range, get a bigger one or a smaller one. The melody can move from one player to the other. That's the way early compositions for ensembles worked.
It seems very easy to forget how we arrived where we currently are with musical technology. Once upon a time, our ears were the calibrating tools. Ponder the question of how to produce and manufacture an accurate tuning reference in the mid 1600s. Technology was not there, but the perfect math of harmony was there, as well as the common ability to hear it. Ask for a pitch reference and someone would probably sing a note for you. Then all the following notes are harmonically related to the first one. How else are you going to find them? So our ears derived the math raw because that was the experience of it;- raw air vibrations in our head, and our ears and head were up to the job. Then we used our ears and head to invent technology and now that invented technology has become the tail wagging the dog.
When in doubt, tune to the oboe. If you are on a budget, tune to the recorder.