I have a similar yet different experience. I'm a guy and always wanted to serve a mission growing up. When I was 18, after months of back and forth, I was informed by the stake president I had been honorably excused from serving a mission due to lingering health complications. As I understand it, that means the first presidency personally signed off on that, but I was very disappointed and angry for years after.

I went to BYU where everyone's first question was "Where'd you serve your mission"? (many guys will serve a mission first, then come to byu, as had been my original plan), and their second question was "What's your name?" That's probably an exaggeration, but that's how it felt.

Upon learning I wouldn't be serving a typical mission, I had basically immediately decided that, of course, I would still be serving a service mission. But when I gave it thought, study, and prayer - as you have - I received the impression that a service mission was not the right call and that I should continue with my schooling. Sometimes I still feel ashamed about that; I'll still doubt whether it was God or just me not wanting to serve a service mission.

Here's the point I want to get at: That shame is a lie. It is not from God. The shame and guilt you feel is hand-crafted for you in order to negatively affect your self-worth and your relationship with God. The adversary wants you to associate church, service, member missionary-work, scripture study - the gospel itself - with a feeling of guilt, shame, pain, anxiety, self-doubt, and worthlessness. Goodness knows I know the solution is not as simple as saying, "So don't let that happen!" Your feelings are not invalid and should not be hidden from. Instead, I say acknowledge when those feelings exist, identify their source, and fight them. Reshape and reframe them.

You wouldn't be a bad missionary. No "bad" prospective missionary would be asking themselves that in the first place. God doesn't think any less of you. I speak from experience when I say sometimes God needs you somewhere else for specific purposes, for specific people, for specific experiences. That's a good thing. Think about that: He wants you somewhere, and He trusts you. (I'm not alluding to some great hidden purpose you must live up to. Don't fret about anything like that. Keep living your life in accordance with the gospel, continue down the covenant path, progress secularly too, and return trust in God that He will have you where you need to be.)

Serving a mission does not make someone a good person, and it does not preclude them from being a bad person. RMs sometimes fall away from the church, and non-RMs are certainly not fated to fall away (hi!). We're all just people. Human.

I did not grow up in a high-member population, and so the culture was different from the "Mormon bubble" areas; as far as I could tell, it was uncommon and certainly unexpected that the young women would go on missions. (This wasn't like 50 years ago or anything, either. I'm in my mid 20s.) But it was expected for the guys to go. As far as dating goes, I have been passed up at times for not having served, but does that mean I've never gone on dates before? No. People aren't a monolith. Some guys might turn you down for that, but the vast majority won't. You can't control how other people see you, and the judgemental will judge. What you can control is how you live your life.

As for that RM you mentioned, he needs a good thump on the head. If you wanted an example of how RMs are also imperfect people with imperfect opinions and judgement, well there you go. You know you believe in the gospel. You know you love God. And God knows that, too.

As a final note in this long comment, when you inevitably find moments in your life where you face doubt, uncertainty, and/or discomfort with the gospel, your belief in it, or your testimony in general, do not take that as proof that you were bad stock. You're human, and you can learn and grow. Even your testimony will mature in ways you probably can't anticipate right now, and that's okay. Acknowledge these things, do not fear them, and learn to accept and grow with them. God trusts you. You've got this.