What effect will not paying COBRA for April and not signing up for an ACA marketplace plan (thereby going uncovered for one month) have?
None. Your new employer coverage will begin May 1; and presumably are covered now. You would be responsible for any claims with dates of service in the month of April but that's it.
As to the question in your title, yes. There are plenty of situations where a person with a chronic illness would remain on employer coverage rather than securing individual coverage. Employer plan formularies and doctor networks can be much more broad than individually available plans, and individual plans are more subject to change from year to year; which could disrupt the care of a chronically ill person. In addition to that, it's rare but depending on the person's age, there are cases where COBRA costs are lower than comparable individually available plans when the employer plan is composite rated but individual plans are age rated. While the guarantee issue nature of individual health insurance has radically diminished the utility of the COBRA law there are still cases where a person would chose COBRA even in the face of increased costs.
Not that it's terribly relevant, but COBRA is just a law that says your employer needs to allow you to remain enrolled on its plan for up to 18 months; some states extend this requirement. COBRA is not in and of itself expensive; COBRA is just the gross premium your employer was paying plus a 2% surcharge. Your employer's coverage was expensive and being heavily subsidized.