What benefit (if any) do horses or carriages offer when travelling long distances?

The rules for Travel Pace in the PHB and Basic Rules are good and straightforward, with players able to travel on foot for 8 hours per day without over-exerting themselves, choosing a pace of Slow (2 miles/hour, 18 miles/day; can move stealthily), Normal (3 m/h, 24 m/d) or Fast (4 m/h, 30 m/d, take a penalty to pass Perception).

In this same section, there are two paragraphs about mounts and vehicles; they specify that mounts can gallop at twice the normal fast pace for an hour, and with frequently freshened mounts – typically only available in highly populated areas – a rider can travel fast over long distances. However when it comes to vehicles, only water going vessels or flying mounts get you travelling faster; for commonly available forms of transport:

Characters in wagons, carriages, or other land vehicles choose a pace as normal.

Is this right? Rules as written it seems there is no benefit to travelling by wagon or carriage, and certainly it takes the same amount of time as walking. In my current game of Dragon of Icespire Peak, the players have spent quite a bit of time demanding horses or a carriage in order to travel the 65 miles from Phandalin to Butterskull Ranch, believing it would get them there faster (as the quest there seems urgent). But looking up the travelling rules it really seems there’s no speed benefit to them paying for a ride or hiring horses, except that they might shave an hour off their travel time each day in the latter case. That makes sense for a wagon which is designed only for hauling stuff, not speed, but what about a carriage or a riding horses?

I’d like to know if there are any additional benefits (beyond the purely narrative) that I am missing for travel by carriage, or any ways that mounts or vehicles can decrease their travelling time (beyond the double fast pace for one hour).

I’m also interested in any house rules you may have used to give players a speed boost through riding over walking.