How much of a pivot can you really do from a T15 school?

A bit of background on myself. I've worked in Corporate Strategy in 3 different industries and I am currently a CS Manager in my 3rd job/firm out of college. Honestly, I never had a master career plan and just took on new roles because they paid better and I was pretty agnostic about what industry I worked in.

I am currently applying to a few T10-T20 B Schools mostly because I am burnt out and I need a break, but also would like to transition into either Banking or Consulting post B School. I will have a total of 5 years experience entering B School.

Now, I am wondering how much a of a transition can you really do out of a T15 type B School. I was talking to a McCOmbs alum doing BB banking in Houston and he said I would have a really hard time breaking into banking in Houston because nothing in my Resume shows that I am interested in energy banking. I was probably naive in my thinking, but I thought you could use B School to pivot into a new career. This guy basically told me I would have to recruit for the banking group focusing on the 3 industries I have worked in.

So my question regarding recruiting for banking/consulting out of B School are these:

1.) Do banks care about what industries you worked in when they are recruiting for Associate positions? Do you have to have Healthcare exp to recruit for HC banking or have to have O&G work experience to recruit for O&G banking, even out of McCombs or Rice?

2.) Do Consulting firms care about what industries you worked in? MBB/T2.

Thank you in advance for your responses.