OP- a kid who wants personal relationships with faculty will have them. This is independent of the size of the U, the size of classes, or anything size based. How do you think kids from ginormous U’s get into grad schools where recommendations are critical? They ask their favorite professors, that’s how.
One of my kids closest faculty relationships was with a professor who had never taught a course the kid took. But the kid got a job fact-checking a manuscript, which led to a job editing a paper, and by the time the year ended, they were having coffee and chatting and bouncing ideas off each other. This professor picked up the phone and got the kid a fellowship (complete with $ and travel stipend), this professor said “You need a job after graduation, just let me know how I can help”.
You can take a 14 student seminar and never make eye contact, never go to office hours, and never cultivate a relationship, or you can take a 300 person lecture class and be that kid who shows up for office hours to ask questions, volunteers to turn the professors notes into slides before the review session, and before you know it, voila.