A family of Microsoft relational database management systems designed for ease of use.
Look at Tools, Options, Schedule, AutoLink inserted or moved tasks.
If this is off, turn it on but for more complex projects it is better to have off.
Remember that a computer has an IQ of zero, so if you blindly say automatically link predecessor and Successor if I delete the task, then it will. Sometimes you won't want that to happen and you end up with flawed logic.
I sometimes have this on when developing a schedule, but once under way I always turn it off. I want to know about any and all links that are created and I want to be the only one to do them. So your suggested blindly add links when tasks are deleted would be very unwelcome by most schedulers of complex projects. (Be careful of what you wish for.)
So Project can do what you want, but thankfully that option can be turned off so schedulers can have good control of their links.