You could form relationships with recruiters. You don't need to be close with recruiters, just reliable. I've found that connections with other professionals is better for getting into niche positions, but good recruiters have placed me into excellent positions as well. It's just a slog to weed out crappy shotgun recruiters from quality ones.

Ultimately, though, I ponder about your professional connection problem. You don't need to be super close friends to have people refer you for things. Just pleasant. I see what you say, but part of your career as you advance is managing people's feelings. If people don't walk away feeling good from interactions with you, then I don't think you're doing your job well. That doesn't mean you have to smiley and be fake, but people can't feel like you don't care at all about what they are emotionally invested in whether that be the project or their career or whatever. It's worth considering that you might be lacking a skill for higher levels of work and even if you got those positions you'd hate them because of how much emotional labor is involved.