It’s complicated!
Many schools that have 1st Yr Engineering still have students apply to an intended major, but don’t “officially” have them declare until later, sometimes as late as 2nd semester sophomore year (i.e. Cornell). And, most schools allow flexibility to change engineering majors as long as you’ve been admitted to their college of engineering.
And then there is the whole what is a first engineering class? At some schools it is just a 1 or 2 credit seminar that discusses all the engineering disciplines but there is nothing hands on.
OP - based on what you describe, you may want to look for a school that has a first year engineering design class. My daughter’s freshman design class was project based learning that introduced different disciplines, had guest lectures from various industries, and they took field trips to tour companies.
WPI would be a good school to check out.
Clarkson, as mentioned above, has a first year engineering studies program for undecided students who are interested in engineering but a student could easily pivot to pure math or physics and be on track to graduate.
Tufts is doing a pilot program to introduce different engineering disciplines to freshmen and they have a design challenge day.
I’ll also add that any university will have a career center that can also help students identify the right major for them.