Chemical, electrical, and biomedical engineering are all different fields that take a different set of major specific courses. Engineering companies who hire new grads for engineering specific positions often expect students to learn key skills in these major specific courses, so they often need electrical major and not chemical major or vice versa. For example, I majored in electrical engineering and work in a related field. It would be quite rare (rare enough that I’ve never heard of it happening) for a chemical or biomedical engineer to be considered for a position in my subfield. Instead they’d typically apply for a different type of engineering position that more directly relates to their major.
Cross over between majors is more common in positions outside of engineering. For example, a small portion of both electrical or chemical engineering majors might choose to initially work in finance or consulting, rather than engineering.
The classes required for a chemical engineering major typically fulfill the pre-med lecture course requirements for calculus, chemistry, and physics. University general requirements would likely fulfill English. This leaves biology, which will not be fulfilled by chemical engineering major requirements and will need to be taken as an elective. An intro statistics class would also be a good idea, if not part of the required major classes. Labs also are sometimes not required for chemical engineering major, but are for pre-med. Pre-med is a significant addition beyond a standard chemical engineering major, particularly with using electives for a year of intro biology, but I’d consider it a practical addition.
A chemistry major would likely have similar overlap with pre-med and need to use electives for intro bio, just like the chemical engineering major. The difference is the chemistry major has fewer major specific classes required beyond the pre-med classes than the chemical engineering major.
I expect the rarity of chemical engineering + pre-med primarily relates to typical pre-meds favoring the more common major path, with fewer additional classes required beyond pre-med classes. Pre-meds and engineers also often have different personalities and interests. I doubt that the typical pre-med would want to be an engineer, if med school didn’t work out. The few pre-meds who do choose engineering majors tend to have good outcomes, with a high rate of med school acceptances, as well as high GPA + score stats. It’s been awhile since the AAMC published MCAT scores by major. Back when they did several years ago, the MCAT scores by major were as follows:
Biological Sciences MCAT Score by Major
Biomedical Engineering: 10.6
Electrical Engineering: 10.1
Chemistry: 10.0
Biology: 9.7
Pre-medical: 8.9
Physical Sciences MCAT Score by Major
Biomedical Engineering: 11.1
Electrical Engineering: 10.9
Chemistry: 9.5
Biology: 9.0
Pre-medical: 8.5
Verbal Reasoning MCAT Score by Major
Biomedical Engineering: 9.6
Electrical Engineering: 9.4
Chemistry: 9.0
Biology: 8.7
Pre-medical: 8.1