We speak the same language. Period.
-Shaw
I certainly believe that is the biggest reason. It may not be the only reason, but I think it is the biggest.
The connection between Great Britain & the US runs a little deeper than just saying “we speak the same language”.
Both also have similar cultural outlooks - namely, a respect for the individual, religious tolerance, “acceptance of the eccentric”, and a focus on business and the consumer (the country described as “a nation of shopkeepers” naturally gave birth to “the business of America is business”) that you didn’t see in other countries. Add in a similar common-law-based justice system, similar military structure, similar civil-administrative systems, etc., and the two are natural complements.
But as for Great Britain acting lousy towards Americans in the early 1800s, be assured even GB’s own colonists at the time were ticked off. In Canada, a good example were the naval “press gangs” that roamed colonial Halifax. Because conditions in the Royal Navy were so bad, they had a constant problem with sailors deserting - so to solve the problem, they just created mob gangs of sailors that literally grabbed men off the streets and “pressed” them into naval service. Think about that - you could go out for a nice evening walk in Halifax, then around the corner comes a press gang, you get beat up and kidnapped - when you wake up you’re doing menial duty on some RN ship in the middle of the Atlantic with no chance of escape! No wonder many Canadian colonists wanted to separate.