So first, yeah, super common misconception. BrE doesn't sound like BrE did in the 1770s. AmE didn't "change faster". This is aside from the fact that there is no one single BrE accent, which I only bring up to head off whoever else might show up and complain about it not being addressed.

Second, language is constantly changing. All languages all the time are undergoing changes. These aren't always happening at the same speed, and we certainly can't say "language changes at a rate of X every Y".

Third, yes BrE accents were previously rhotic. A number of other features, such as -ing endings pronounced as -in', were also previously markers of high-class accents.

The problem is we can't really say "why". I mean we can give reasons why such things happen, but most sound changes happen unconsciously. I speak a tiny bit different than my parents, but it has nothing to do with me actively making that effort.

One reason why is that new generations of speakers "mis-hear" the speech of the older generations. Or, to put it another way, what might be an secondary feature gets heard as the primary feature, thus shifting the accent. This is how a lot of changes have come about in language. For example, when you say "pa" vs when you say "ba", your "ba" is probably a little lower in pitch than your "pa" even if you're completely unaware of this.

However, it's entirely possible that a future generation hears this distinction and thinks that's more important than the p/b difference. That's exactly one way in which tones come about in language, and that exact example has happened a number of times throughout history. To you, the pitch difference is nonexistent it's so unimportant. But maybe your kid's generation thinks it's the significant point.

Another way things change is that lower class speakers may emulate upper class speakers. Upper class speakers may notice this, and so gradually shift. It's the same reason Tiffany went from an upper class name to being seen as "trashy" (apologies to any Tiffanies out there). It happens with names all the time, and can happen with language more generally as well.

Language change happens often without much thought to why. Yes, people can make semi-conscious efforts to differentiate their speech from others, but for the most part, the changes are natural and any explanation are typically going to be after the fact, not something people set out to do.

At some point people stopped pronouncing their R's as strongly, and little by little it disappeared over the course of a couple generations.