Done right, an inserted tenon can work just fine. But, screwing around with them at the end of the lower joint of a soprano clarinet is just asking for trouble.
Yes, they can work very good. Actually, the tenon is the realtively weak area of the body and a (properly) glued tenon, preferably reinforced, might actually be a lot stronger than one piece. Though IMO not really worth since the one piece is not a problem and rarely break (at least I almost never see broken ones, including from schools, though I know in some areas people see more of them).
I have been told that this process was responsible for the higher rate of price increase of bass clarinets, as the one foot plus pieces essential to making a wooden bass clarinet have become increasingly rare.
I guess that is possible. In the case of low C bass clarinets even much more than that, about 65cm long for the lower joint. I think Selmer Paris is still making them from one piece, but Buffet now uses two pieces for the lower joint.
But do you know how much of the cost is the wood itself? I think it's very little of the total cost to make a good bass clarinet. Even if they got the wood for free I don't think the price would be that much lower.
I'm a bit unsure why this kind of manufacture was dropped. One would think that it's easier to manufacture. Might be because repairs would be a bit more difficult.
A lot of problems with that. Like you said, repairs are more difficult, not only to do but also to sometimes to find the problem. Not only for a repairer but for the factory when building it (assuming they check).
A major problem, especially without a barrel, is intonation. Many players use barrels in different lengths, or open and close them, depending on tuning. I remember a recording I did in a place almost like a sauna and needed to play 440. I used my longer barrel and needed to open by about 4mm, plus open at the middle. If I had all one piece it would simply be impossible to play in tune in those conditions.
I think a one piece would actually be harder to make. Although no need for accurate tenons, corking, I think all this is done pretty easily, at least by the big companies, on a CNC lathe and assembly line style probably. I remember one person mentioning he went on a tour at an American wood manufacturer's factory. They happened to be making one piece (plastic) bass clarinets at the time. They were drilling tone holes with the body mounted on both ends, it flexed so much and they didn't do anything about it!