I think you're overlooking a very important qualifier- he doesn't say "We'll be there in two weeks". He says "We have at least a fortnight's journey ahead of us". We also don't know whether Aragorn would normally use the roads: as a Ranger, committed to secrecy, he probably uses roads as seldom as possible.
He also drives the hobbits very hard on the trip: he is constantly rushing them along, and shows very little sympathy for their hardship in trying to keep up with him. When Frodo is wounded, we might expect Aragorn to be more sympathetic, but he can't afford to be; now, more than ever, time is of the essence.
The hypothetical timeline Aragorn mentions - that he would be able to cover the distance in 12 days - also doesn't take into account that now they are being pursued by the Nazgûl. Although it is likely that Aragorn frequently has important business to deal with, he has never been entrusted with a task so vital for the survival of the world.
All of these factors - the importance of keeping the Ring safe; Frodo's injuries; the Nazgûl being hot on their tails - combine to make haste an essential concern. They don't have time to tarry. They must move more quickly than they ever have before. Delay would mean Frodo dies, the Nazgûl take the Ring, and the world is destroyed. Aragorn sets a demanding pace because he has no other option.
