Oh Prettyfly, you are not weak at all. I went through this with my second child and I was absolutely terrified of going into labour on my own. But at the same time, I really couldn't think of anyone I wanted to be present. My mum, sister, best friend, all have very specific relationships with me, but really not birth partner stuff, as far as I was concerned.
In the end, I had to have an(other) elective caesarean, so didn't have the problem of labour, and my sister was with me for the op - very different from labour, I think.
After they'd all gone home and it was the next day and I saw other mother's visitors coming, yes I felt shit. When a proud father walked in with a camcorder, I cried for about an hour and a half. I'd deliberately ordered one of those private rooms (if you have a caesarean you get priority) to avoid stuff like that.
None of that changed the fact that what I was doing was right.
And also, bless them, the midwives were so overwhelmingly kind to me. Maybe it's just the hospital I went to, but perhaps if you're on your own they are extra nice. (They were certainly nicer than the ones I had with my first child, when xp was still around - but that was in a different hospital, so maybe they were just nicer anyway!)
I'm not trying to depress you, just to let you know that what you're feeling is absolutely normal and however many tears you cry in the hospital (and there probably will be a few) you will get through it. And you won't be a crap Mum - the very fact that you're thinking about it now shows how seriously you're already taking the job!
Best of luck to you.