Were the rejections from REU’s? if so, that does not necessarily indicate how grad school admissions will work: they are very competitive and they tend to choose students whose background is the closest fit. For example Collegekid2 was told that she got an REU b/c of her familiarity with a specific piece of equipment - they wouldn’t have to ‘waste’ time training her to use it! So, it can be that random.
However: getting some form of research IS crucial. It’s fine if it’s at the home university, but a PhD is a research degree and they want to know that you are ready to be part of a research program for 4-6 years.
Not your question, but this:
is problematic. There is no “we” here- it is your student. Whatever else, a parent should NEVER make contact with the career development office at the university OR with a grad program.