Boeing doesn’t make engines. Also B737’s use the same engines as the Airbus A320 family (with minor differences for fitment)
They’re either CFM56’s in older A320’s and B737’s or CFM LEAP’s in the latest A320 NEO’s and B737 MAX’s.
Those engines are made by a subsidiary of GE and a French company. Very reliable.
You’re right, of course. And the maintenance would be the responsibility of the airline.