The Dangers of Travel
While we tend to bemoan how dangers cars are, traveling by coach had a lot more issues and a lot less safety features. Getting into a coach was no guarantee that you'd get there safely for a number of different reasons.
First off, there were two terms for robbers: highwaymen and footpads. The first robbed on horseback, while the second did so on foot; carriages often had to stop for various reasons, and the footpads could simply walk up with a pistol and rob them -- or stand in the road and make the horses rear and stop. We tend to think of footpads as only robbing pedestrians, but they were known to "carjack" as well. In fact, because horses were expensive to own and maintain, they tended to be more common than highwaymen.
While it's popular in modern depictions of the 1800s to show poor travelers to be assaulted by highwaymen every other day, in reality, the Regency era didn't have a huge problem with such crimes. In the early to mid-1700s, it was a serious problem, but between travelers protecting themselves, high bounties on robbers, and a serious legal crackdown on offenders, the problem was mostly eradicated by the 19th century.
According to "Coaching, with Anecdotes of the Road" by Lord William Pitt Lennox (which is a first-hand account of coaching throughout the 1800s), Lord Lennox states that robberies continued until about 1814, which was the last account of a stagecoach being robbed. However, even before that, it wasn't a common problem. He states that in all the time he spent traveling (and he did a lot of it), he was only robbed twice. So, really it was a problem, but not a common one.
The crazy thing is that robbery didn't always happen just on long-distance trips. Lord Lennox also stated that up to the 1750s, most robberies happened on the streets of London. Of course, the demand for better public safety led to the government stepping in and going after the problem.
Carriages were way more dangerous to drive than cars. Between the fact that they were top-heavy and difficult to maneuver, horses could be dangerous and unpredictable. In 1866, an average of four people were killed by horses and carriages every week in London. To put that into perspective, modern London has eight times more people and roughly thirteen times more cars than all of Great Britain in the 1850s. Yet in 2016, London averaged only two and a half deaths per week. So, modern London has significantly more people and vehicles on the road with nearly half the fatalities. So, yeah. Carriages weren't very safe.
One common issue was the fact that many people simply didn't know how to drive them properly. Lord Lennox doesn't give a very sterling recommendation for coachmen in his time. Throughout his book, he paints most stagecoach drivers as drunk and unscrupulous men.
The opening of Honor and Redemption was directly influenced by Lord Lennox's book. He said that most young men would learn how bribe coachmen to allow them to drive. Driving was very popular and driving a "four-in-hand" was considered the peak of coolness, but it wasn't always easy to get the opportunity to practice. So, the young men seized the opportunity when traveling to and from university. They'd offer the driver a few coins and be allowed to drive the stagecoach themselves. Of course, they didn't have a clue what they were doing and handling four horses at one time was not an easy thing; driving a carriage isn't like driving a car -- unless your car has four engines that all have minds of their own and a tendency to be spooked.
Needless to say, this led to a lot of accidents on the highways and toll roads. It makes me think that if a family had a professional coachman on staff, that would have to be a very trusted servant. You are literally putting yourself and your family's safety in his hands, and that is not something people do lightly.
But even if it wasn't pure idiocy on the driver's part, carriages just weren't super safe. Governess carts were introduced in the early 1900s as a safer alternative to the more popular dogcart, which had a bad reputation for children falling out of them. Two-wheeled carriages had a high center of gravity, making them especially easy to tip. Well-trained horses can be very reliable, but a horse is still a horse and can be spooked. Road conditions were terrible, leading to carriages slipping off the road and tipping. So, not a great recipe for safety while traveling.