Connection Tamagotchis do not die of old age, instead they die of care mistakes, basically when one of their meters completely empty. After the matchmaker stops coming, the tamagotchi will evolve into a senior and keep on living till it dies or marries another senior to have children. Unless you have another connection Tamagotchi, there is no way to start a new generation with an oldie.
I too like the limited lifespan feature of the vintage Tamagotchis as it makes them seem more alive. Since I have both the mini and a few vintages, I can give some more depth on them. In general, the only differences a vintage Tamagotchi (pre-connection models that started in 1996) has from the other versions is that its growth is dependent on discipline and care mistakes and it is raised from birth to death. Discipline only comes into affect in the toddler and teen stages and is indicated when the Tamagotchi beeps but its meters are not empty. Throughout its life it only requires being fed, being played with, having its messes cleaned up, being cured when sick, and having its lights turned out when it is asleep. When it become an adult, it only requires its basic needs and as it grows older, it become needier till it dies from care mistakes or its lifespan has a set number of days. The adult stage is basically the past stages without the discipline and as nothing but sickness and eventual death will happen, it can become unexciting. But in the adult stage a vintage tamagotchi probably has a week of life left, so it may not become noticeably boring. There is nothing to do with an adult vintage tama but care for it, unlike the other versions that have purchasable items, seasonal events, or the ability to visit other tamagotchis. The P1 and P2 are basic vintage Tamagotchis as the later versions are themed and have their own growth quirks.
The mini is a truncated version of the original vintages, the P1 and P2. The mini only has two growth stages: baby and adult. The baby stage is not needy and lasts a whole day. Growth is determined solely on care mistakes as there is no discipline. The features removed from the mini are simply discipline and the toddler and teenager stages. Basically, the mini is like a vintage tamagotchi with an extended adult stage which, as I have expressed, can become boring because there are no other features. One thing worth mentioning is that there are secret character on all vintages and the mini and these characters evolve from adult characters, in a sense being an extra stage. On the mini, secret characters are easier to obtain as they simply require good care.
Jhud has recommended you the earlier connection Tamagotchis, which may be a better option if you want something more interactive than what I have described about the vintage Tamagotchis and the mini. The only thing with the connection Tamagotchis is that the adult stage really only lasts 3-4 days as after the matchmaker stops coming - which I think she does consistently for 3 days - then your tamagotchi will evolve into an oldie and be forced to keep living. That is my only gripe about the connection Tamagotchis with how the adult stage is reduced unnecessarily. But there is one way around this. The Japanese connection Tamagotchis have a longer adult stage because the matchmaker comes later; however, the character pool on the Japanese connections is smaller than on the English versions. But if you like or don't mind the characters, it might be worthwhile to get a Japanese connection. Additionally, the language element is not difficult as the Tamagotchis have nearly the same functions, just different labels.