This is an important question because the answer can illuminate college incentives for enhancing student completion. At some private colleges it costs a lot to recruit a new student. So helping academically struggling students can make sense economically. There are costs to the personnel needed such as admissions officers and scholarsip money that is wasted on drop outs.
But for open access schools with many students who want to get in, there are minimal recruitment costs , but large costs for remediation and counseling. So , it could be much less expensive to obtain a new student than pour resources into an existing student. This leads to student churn that preserves the full time enrollment at a college. Most states pay based on FTE.
More research is needed on this institutional incentive issue, but I suspect there are many cases where the answer to my headline question is a resounding – YES it is less expensive for the college to recruit a new student who is waiting to get in.