Engaging other players is about good communication, and good communication is [i]everyone's[/i] responsibility. I've played both the "outsider" who is alone at the start of the story and the social butterfly who makes as many connections as possible. How effective each approach is depends on their execution as well as the type of RP it is. 

One of the weakest ways to get other people to engage your character is to enter a crowd and have your guy/gal shout "Hello everyone! I'm [i]*insert name here*[/i], how are you all doing?" While it's clearly an open invitation for anyone to approach your character, it's not personal enough. It gives other players very little to work with. Not only that, but how many times have you seen someone do this in a group of strangers? Few people have that kind of ego and/or are that out of touch with proper etiquette. You need to be a bit more selective. That's not to say you couldn't open up with something like that, but then if your character is a womanizer, maybe then have them approach a female player character and be all like: "And you especially, I think we might have a few things in common." followed by some suggestive brow movements. Now the character you have approached has some material to work with. They can react cautiously around your character, or maybe they enjoy the attention and continue to converse. Maybe this bothers another character who thinks your character is being rude, and says their piece. Then maybe another character reacts, and before long everyone's taken notice of your character. Alternatively, the character your character just hit on might choose to ignore you in favor of someone and something else. That's fine, there are plenty of other players to engage with. 

For a loner character, it's basically knowing when to join the story. Which should be as soon as possible. Just keep an eye out for an opportunity. The GM may push you along, but it's not their responsibility to give you special treatment. I would much rather join the story on my own terms before being forced into it. What Penny said holds true though. Some people just want to play in a setting without doing anything with the story.

You'll also be a lot more successful if you get to know the players. Pay attention to their antics OOC/on discord and you'll know if they are interested in being interacted with. The silent guy who never comments? They might need some more time to establish their character or to get invested in the player base. The other one who occasionally comments and is either ignored or is swept aside by some of the more vocal members? Ripe to engage with. There's also no harm in asking if anyone would like to write with you. We are role playing for collaborative adventures.

Loner players might just want some time to set up their characters or want to be outsiders, and might not be engaging because they aren't ready. Other times it's their lack of experience or inability to compromise that keeps them distant. Again, it's really easy to solve these issues by just [i]talking[/i] to people. 

Though in really big roleplays, I would argue that people who want to have their character know and be friends with everyone can be just as irksome as people who want to stay solo. I mean we all want a big ol' anime harem, but come on, even if you like all the players, your character can't like all the characters!

I guess the short version of all this is:

-Communicate, communicate, communicate!

-Players genuinely prefer personalized experiences. "Hey y'all!" will not get a reaction as strong as "Hey, Cindy!"

-The better you know your players, the easier it will be to engage with them. 

-Loners probably plan on joining the story later, or might need some help.

-You can't waifu them all, don't even try.