Generally speaking, any university with a graduate theatre program offering MFAs will have a large and robust theatre department with plenty of classes and opportunities for undergrad involvement, particularly on the tech side. Look, for example, at the Universities of Arizona, Washington, and Oregon. If you have grad students pursuing MFAs in design (costume, scene, lighting) there is always a need for students to help with the production end of things, whether you’re a major, a minor, or an enthusiastic non-major. Most BA degrees with a theatre major will allow you to get a broad background in the field, yet still spend lots of time in your area of interest (technical theatre, in your case).
Many years ago I double-majored (theatre and history) at a state flagship. By the time I graduated I had gained a ton of experience, had developed a decent portfolio of design and tech work, and went on to an MFA in design. I subsequently taught in smaller private and LAC theatre departments which, while good, were a little less lively than their large state counterparts. That’s not necessarily a bad thing, as smaller schools/programs have their own advantages and suit some students better – it’s just the nature of the size of the departments and the scope of their programs. As someone interested in technical theatre, you will likely find opportunities wherever you end up. Best of luck to you!