I only play Argonians, and all of them embrace their Dragonborn status. As to why they're there, well, that depends on the individual. One was there because she was an Imperial Agent on an undercover mission; when she got caught she couldn't use her influence to escape without blowing her cover (nobody was supposed to know why she was there, save those who sent her), so she was sent off with the rest of the prisoners.
Another was a Thief who'd made Cyrodill too hot to hold him, so fled northward after hearing there was a similar organization there in need of help. He got caught in the ambush on the way back to Riften from a job in Falkreath.
The current one is a mage, who disliked the narrow-minded attitudes of some of the Mages' Guild leadership and headed north after hearing that the College was more accepting of 'free' thinkers. As she'd been warned that the Nords distrusted mages (to say the least), she was keeping a low profile by only using her magic while alone; as such, when caught in the ambush (she was just passing through) she chose not to fight so as to avoid 'outing' herself and causing unwanted problems.
In all these cases the character him/her self does not know what's in store at Helgen, so does not know that it would have been better to risk an escape attempt. As we know, it's not going to matter, but the characters themselves don't/won't until they're already there and about to die.
For me, at least, there's a bonus while in Helgen if playing an Argonian- when the Captain calls out "Next! The Lizard!", Alduin's roar that follows almost immediately comes across as him saying "Who you callin' a lizard, lunchmeat?"
That last line has me laughing!

Lots of good advice here. Anyone can be Dragonborn. Race is not an issue. But as others have pointed out, you can just ignore the main quest and never see a dragon or learn a shout. Just do not deliver the dragonstone to Farengar. Delivering the stone trips off the fight with the second dragon. You can even clean out Bleak Falls, just don't pick up the dragonstone, and you will never see a dragon for the rest of the game. Then you can be just your average Joegonian in Skyrim.
In addition to the many other ideas, some possible RPs are that your character is hunting a Dark Elf who fled there after the Red Year. Said Dark Elf may be some old enemy of your family, who enslaved and tortured your ancestors, etc... Now it is time for revenge. Or you could be a Shadowscale assassin, pursuing a mark into Skyrim. Or maybe you are an unusual Argonian, who dislikes the heat, and loves the cold? A Frost-Gonian if you will. This may even be the result of magical experimentation. Skyrim would be an obvious destination then.