Rocket engines are very complex machinery that's difficult and expensive to make. Reusing the rocket allows you to reuse the engine.
SpaceX has also managed to show that reused hardware can be more reliable than brand new hardware. You run the hardware through a number of tests before launch, but there is no better test than an actual launch. Satellites still cost a lot more than rockets, so reliability is a big deal
ULA calls this "SMART reuse" but has not yet successfully achieved it with their Vulcan rocket.
https://planetary.s3.amazonaws.com/web/assets/pictures/_2400...
In some ways this is simpler, but in others it is more complex since you need a way to catch the engines before they dunk in the salt water. A rocket booster that is already set up with avionics, engines, and fuel can guide and land itself without needing a precise helicopter catch etc.
I'm not sure how leaving the rest of the booster would make a propulsive landing easier. It seems like more complexity than just leaving some fuel in the tank, for less benefit. But ULA's Vulcan (which incidentally uses the same BE-4 engine as New Glenn) plans on using a helicopter to catch it's engines as they parachute down [1]
This concept is generally known as "Stage-and-a-half", and it was used on the Atlas rockets
As Diane Rehm once lamented on her radio show about two decades ago, as she was interviewing some cutting-edge engineer about his work, “why does it have to be so haaaard?!?”
I do wonder if there's a way to reuse engines without reusing whole rockets, that could wind up cheaper. Pop the engine out of the rocket with some small fuel tanks, spin round and do a retro burn, then ... somehow land. I have not worked out all the details.