There are ways to make deeper relationships faster. The key is to become an irreplaceable existence in each others' lives. People fall in love and become closer than any childhood friends (in several aspects at least) even if they meet late in life, right? You basically need to create that kind of relationship with whoever you choose to cultivate a deep friendship with.
You can certainly overcome the time condition with practice on getting closer faster. I spent 2 years as a nomad traveling through and living in ~8 vastly different cultures where I didn't even speak the language and had no local friends before I got there. Before that, I was born in India, grew up in the Middle East, studied on the East Coast, and worked in California. During all these transitions I went through lots of periods of incredible loneliness and missing out on the long-standing relationships others had, which is I guess how I learned most of this. Some general tips based on that experience.
- Cut the chitchat. Skip the "where do you work? where did you grow up? what's your favorite color?" bullshit when you meet someone. It isn't enriching, and it's more often than not forgettable for both people. You will learn these things about each other over time if you become closer, so save it for later. Don't initiate these questions, and find ways to segway out of this into the other stuff (see below) if the other person initiates it.
- Forge a mentor/padawan relationship. If you know more about something they're interested in, or vice versa, go for this, and do it in a respectful manner whichever side you're on, but also be willing to treat them as a peer. This instantly makes you irreplaceable as a friend they can and want to learn from (or teach to), when it's something they're embarrassed to do with their other friends.
- Be a good conversationalist. Listen to what they're saying, ask interesting questions, always prioritizing making them feel comfortable telling you more stuff and think more, rather than just making smalltalk. Try to relate to what they're saying but with as little talking about yourself. Direct the conversation towards interesting aspects of the topic rather than mundane superficial details. E.g.
Them: Ugh rough day at work
You: Good challenges or stuff you wish you didn't have to deal with? (rather than "tell me about it")
Them: I guess good challenges, but I wouldn't want to face this too often. We were having a crisis deciding which one the core feature of our new product is, and I and the director of product had somewhat conflicting opinions.
(rather than what he would have said "oh, argument with my boss")
This provokes thought, and they associate talking to you with having more meaningful conversations, rather than just regurgitating the same words they'd used with their mom or the bartender who said they looked down.
- Practice under semi-artificial constraints. As someone else suggested, couchsurfing (or living in dorm-style AirBnBs). You will meet other solo travelers or hosts who would like to get to know you. Practice these with them. The time-limit on when you'll depart will encourage everyone to have more fun with you before you go.
- Break down your walls. Be open. Talk about deep topics and expose your flaws and vulnerabilities and things you care about. Talk about things like this, your loneliness and jealousy over other people having deeper friendships. You make yourself vulnerable, but it'll appeal to human empathy, and people will want to help [NOTE: Don't get needy or whiny. Just explore your internal workings together with people]. This draws them in faster and they will want to understand you at a deeper level. They don't see you as that guy who's sitting across the table from them, but as a fellow human being who is in need of their company. People love feeling needed.
- Live together, travel together. Don't live alone. When traveling, don't get a hotel room, go dorm-style. When renting, get a house with a shared kitchen and living room. Carpool. Do a sport with your housemates, work on home decor together (furniture shopping, etc). Go hang out in their room.
- Play games. Sports are not always easy to coordinate, but football/basketball/tennis with colleagues or other friends is a great way to get closer. If those aren't feasible, play video games together. Have a video game party evening now and then for console people, or just play online while voice chatting in a Hangout/Skype call. Make non-game-related conversation when between games. Or if you want to be away from the computer, play board games. With your housemates, a nearby board game meetup, coworkers, etc. Gaming brings out a lot of personality aspects of people, and they don't waste energy on chitchat.
- The above can be summarized as don't just talk to people, but have experiences together. Suffer something together (sport/school/gym), build something together (work/cooking/housekeeping), enjoy something together (concert/games).
The part I didn't talk about much is how to meet people. I'll leave that to you. I hope this gives you an idea on what aspects of your human interaction to improve upon so that you can make the most of the pool of people you do get to meet, and get deeper and closer friendships overcoming the aspect of duration.