Not speed, but distance
As other answers and comments have noted, and as real life history attests, humans trained for marching, under appropriate conditions, can travel just as far in a day as can humans mounted. The advantage of having a mount is in carrying more weight than one could haul oneself at any given speed.
In OP's example, then, is there no benefit to having mounts or wagons if you really, really need to get somewhere fast? Perhaps, if you consider forced marches and endurance.
In the Travel Pace section of the PHB, just above that of "mounts and vehicles" is the part about making a forced march:
The Travel Pace table assumes that characters travel for 8 hours in day. They can push on beyond that limit, at the risk of exhaustion.
For each additional hour of travel beyond 8 hours, the characters cover the distance shown in the Hour column for their pace, and each character must make a Constitution saving throw at the end of the hour. The DC is 10 + 1 for each hour past 8 hours. On a failed saving throw, a character suffers one level of exhaustion.
If the party is willing to continue past eight hours and risk exhaustion, they won't travel faster, but they could travel farther in one day, and arrive sooner than they would have, had they stopped to rest. The first level of exhaustion is disadvantage on skill checks, which might bother the rogue, but which won't deter our party which is determined to arrive sooner 'at any cost'. The second level of exhaustion halves their speed, which will be a serious problem. But even if one of the party hits this after just two extra hours of travel, they will have gone an extra 6 miles at a Normal pace, or 8 miles at a Fast pace, making it more likely they will get to their destination sooner.
Mounts or draft animals can be part of this 'forced march strategy'.
The question then becomes whether characters who are mounted or in vehicles are required to make exhaustion saves along with the animals. In a rules sense, the PCs are still 'traveling', even if not under their own power. In a verisimilitude sense, riding is certainly an activity and is potentially exhausting. However, the amount of exertion one needs to travel while riding depends on the speed of the horse, and if the horse is mostly walking the rider is potentially exerting themselves less than if they were the one walking. And clearly riding in a cart or wagon is less strenuous than walking - some might even claim that that it is possible to rest in a wagon.
As the DM in this case, OP has a number of options on how they rule exhaustion if they want to support the engagement of the players in the world, and their in-game efforts to respond to the urgency of the quest by obtaining mounts or draft animals and getting to Butterskull Ranch 'quickly, at any cost'. The basic strategy would be to have the mounts travel their eight hours and then begin making forced march saves against exhaustion. At the point in time when a mount would drop to half speed, the party would then abandon the mount, redistribute the gear carried to keep everyone underencumbered, and the mountless PC would continue on foot, with their own exhaustion saves somehow favored for having been previously able to use the mount.
Modding the Forced March Rules
Saying that mounted travel doesn't count as travel at all is likely too generous to the PC's, but it might be fair to count two hours of mounted movement as an hour of travel, or four hours of mounted movement as three hours of travel, against their daily limit of eight hours. That way the advantage of bringing mounts is that even if a mount began to founder shortly after eight hours of travel, the PC would be more fresh and could continue to walk for a bit before having to make endurance saves.
Circumstantial bonus
Even for a DM that wanted to count every hour of mounted travel as an hour of travel, for both mount and rider, the DM is RAW empowered to make checks and saves subject to advantage and disadvantage depending on circumstance. Thus they could have the rider's exhaustion saves be made at advantage while they were riding. The advantage of the party equipping themselves with horses, then, is in potentially extending the distance the party can travel before any of the PCs become exhausted.
Redundancy
Even if the DM has all travelers, animals and PCs alike, making all exhaustion saves unmodified, there still is benefit in the redundancy of a mount-rider pair. This is because (assuming the party will abandon exhausted mounts), it will take four failed saves to reduce a character's progress. If the mount founders first, the PC can keep walking. If the PC founders first, their reduction in speed won't matter so long as the mount is still moving.
Wagons keep going
Another option for the party would be to purchase a wagon or cart and multiple draft animals (in addition to their mounts). Walking PCs who reach exhaustion level 2 could be placed in the wagon, with it drawn by the least-exhausted draft animals. The party then could keep going. When the mounts founder, the PCs walk. When the walkers founder, they go in the wagon. The party's progress would only come to an end if either all of their draft animals were exhausted or the weight of exhausted PCs in the wagon became more than the draft animals could pull.
