The other option would have been abandon the staircase entirely and make a taller plane, this would allow a bigger engine with no handling changes, but although

The other option would have been abandon the staircase entirely and make a taller plane, this would allow a bigger engine with no handling changes, but although this would remain a 737 from handling perspective, it would not be a 737 from the airports perspective: it would need to remodel the airports to install bridges or purchase of ladder trucks. No that would not be the major problem. Specifically since practically any airport nowadays has the infrastructure to service such a plane. This was quite different 50 years ago. The reason why they couldn't have heightened it was that a new type certification would have been required and Boeing wanted to avoid exactly that at all cost. Even at the cost of 346 dead people.