Does an old tire become tight? I'm asking this question (see the title) because I wasted so much time/energy (after having a punctured tube) trying to remove the tire and to put it again, while the guys I see on Youtube don't seem to face the same difficulty when replacing a tube.
And at the end, when the tire &amp; the new tube were in place on the rim, the tube started leaking air again.

The tire in question is clearly worn from the outside even though from the inside there is no visible hole.
Do you think if I replace the tire, I wouldn't have a hard time again to stretch the tire on the rim when I have to replace the tube?

P.S: The wheel is a 700x32C (Continental City Ride).
Certainly, old tires get stiffer as the rubber dries out, and this makes them harder to remove and install.

But what you've probably experienced is the difference between brands and styles.  Some tires are a hair oversize and slip onto the rim quite easily, while others are a hair undersize and are much more difficult to deal with.  It's hard to predict which you will encounter unless you are familiar with the brands involved.

(The leak you experienced after replacing the tube was probably due to damaging the tube while installing it.  Unless you really have to, DO NOT use screwdrivers to lever the tire off/on, but use plastic "tire irons", or, better, a "Quick Stick" tire lever.  Using screwdrivers frequently causes tube damage.)