Yup, most engine manufacture is undertaken by specialists. As a good example, Rolls-Royce is an airplane engine company that sometimes makes cars as publicity stunts.
HOWEVER.
It’s not just about who makes engines. Aircraft are meant to last a good 30 years in service and you can’t just ask some schmuck to clean it for $7.50 an hour and call that maintenance. Maintenance is extremely skilled work that tends to be operating under horrible time crunches, especially if a part is suspect and needs to have a plane partially taken apart so it can be changed for a fresh one- a plane that might be due to fly again tomorrow. Maintenance that needs that sort of knowledge tends to have some involvement with the parent company who built the planes, or is even contract work for them.
Boeing is rather notorious for being willing to put the schedule before safety- we’ve seen that in a lot of other accidents. I would absolutely believe that a Boeing manager skimped on engine maintenance because someone in the chain of command said “Get that plane out of the maintenance hangar today or you’re fired, and damn the safety regulations.”
But, that’s just my industry knowledge. The actual circumstances could be way different, so let’s go read the article.