I have done this trip twice (once in each direction) and both times it was one of the highlights of my trip to Japan that year. Fabulous train rides that I will remember for the rest of my life. That said, it was fairly time-consuming to plan all of it, and I stressed considerably over whether I could even get reservations. And, I might say, for this particular series of JR trains, Hyperdia is not a particularly good friend. It will give you most of the information you need, most of the time. You should not rely on it entirely, though. Presumably you have accessed the JR site and read all the details including timetables (I ended up using the JR timetables more than Hyperdia):https://www.jrkyushu.co.jp/english/train/train_index.html
One thing that was a big issue for me in 2018 was that at the time of year I was visiting, Hayato no Kaze was only running on weekends (there are some other options for Yoshimatsu-Kagoshima, but they are not very desirable). So I was limited to days when the other two trains were likely to be heavily booked. Furthermore, I was using a nationwide JR pass, so I could not book online 30 days in advance as I believe you can (?) with the JR Kyushu passes. At any rate, on my trip that year I booked the trains as soon as I got to Japan (for a June date about 2 weeks later), and I managed to get reservations on all three trains (Kawasemi Yamasemi, Isaburo-Shinpei, Hayato no Kaze), but I was prepared with several possible itineraries to allow for one or more of the trains to be full at some times (and they were, some of them). I did not get my first choice of itineraries, but I got one that worked. Note that if you are going during a period when Hayato no Kaze runs daily and you can go on a weekday, things will likely be easier. And there are plenty of times when you won't have any trouble getting tickets.
Of course, that doesn't answer your original question! I would say not to worry about the connection times. Yes, these trains do usually run on time. And JR knows that if they allow one link in the chain to depart before the other link arrives, they will have a lot of very unhappy tourists on their hands. I expect they would probably hold the departure up, within reason. But if you did miss a connection, you could be in for a very long wait somewhere. You would find your way home eventually. Again, I wouldn't worry about that, though.
If you are planning to do all this with a JR Kyushu pass and can book online in advance, I would recommend that. But in any case, I'd say that in the end the trip is well worth the trouble it takes to get the tickets. The trains are touristy for sure (I suppose some ultra-purist rail fans might turn up their noses at them), but I found them to be absolutely delightful, and the scenery is fabulous. Good bentos for sale along the way, too. (I loved the one at Kareigawa.)