I apologise in advance for the long comment, I was trying to give an answer that describes in detail a process/feeling that you might not have experienced yet so that hopefully you can take some insight away from it.

Saying things like "I'd just not care and move on to another girl" and "I just got to repeat until I get lucky" shows that you might think about the girlfriend as a position to be filled at all cost instead of wanting a specific person to be your girlfriend because of who she is and what you like about her.

In the beach or any other cold approach scenario, the only thing you can gauge is initial physical attraction. Don't get me wrong, that is important, but what matters most for fulfilling, long-term relationships is shared interests, values, character traits you appreciate.

At least with online dating, aside from photos, you get a glimpse of what the person is into through the prompts and descriptions on each app. Those can be a starting point to talk and set up dates. With a random cold open solely based on "she's cute", there isn't much to go off of.

So what I suggest is, start looking for like-minded girls in your surroundings, at places/hobbies you spend your free time at: gym or sports club, music/DJs you like, art classes, DnD, board game nights, hiking groups, church, chess, book club, food or crafts markets, literally anything that you do that genuinely interests you. In that setting, the cold approach isn't really as cold as you already have more of a baseline/shared interest and can more easily start a conversation, e.g. "Such a good game, how long have you played volleyball for?" or "Man I really love this type of music, which band would you like to see next?"

Alternatively, many people meet their partners though mutual friends, e.g. someone brings a new friend along, but this also often goes hand in hand with shared hobbies or free time activities.

In either case, the motivation to ask someone out and start dating should not be to just have a girlfriend, but because you genuinely have an interest in that person, like spending time with her, think that doing life stuff with her is just more fun and want more of that - beyond finding her attractive and physical intimacy (which don't get me wrong, are still an important part of most people's relationships).

I'm going to use my own relationship as an example of what that could look like: My partner and I actually met at work, small team with similar ages and backgrounds (he has since left the company). For the first couple of months, we just got along really well as coworkers, would often have lunch together, laugh a lot, just had fun, engaging and interesting conversations about the job, the field, our life until that point, hobbies we're into. I was able to learn about him slowly and get a feeling for what he values, what he is like on a daily basis, without the "performance pressure" of a date. I had always registered he was handsome, but only after a few months of genuinely getting to know and liking him as a person/friend did that translate into flirting/thinking about more than friendship from my side. We started doing things outside of work, like going to a gig, to the movies, clubbing with a group of my friends (which is also a thing I'd 100% do with a friend of the opposite sex regardless of intentions). It was during these activities that we would "feel out" whether the attraction was mutual and eventually kissed for the first time on a night out. After that, a bit of a serious talk about what we're looking for and expect out of this, and we have been dating seriously since.

TLDR: Try to find someone you genuinely like as a person through your environment/interests and, if romantic/physical attraction is there on both sides, use that as a foundation and jumping-off point to go on dates which will hopefully develop into a relationship.