Introduction
The classic Western tracker, more commonly referred to as a bounty hunter, has deep historical roots. To set up the discussion of their effectiveness and answer the second part of your question, it's worth understanding the backstory which led to the rise of private law enforcement which I detail in Part II. However, the primary question seems to be accuracy of their portrayal in media and the methods they utilized which I'll address in Part I.
Part I - Bounty Hunting 101
To begin, I would highly recommend you read Stuart Traub's Rewards, Bounty Hunting, and Criminal Justice in the West: 1865-1900. Despite a dearth of literature describing the legal basis of private law enforcement and/or retelling the lives of famous bounty hunters, very little scholarship attempts a broader historical analysis of the Wild West. If something I say seems to lack citations, you should probably refer to Traub's analysis.
With that said, let's get the most important clarification out of the way. Bounty hunters are almost always not portrayed accurately in movies. Most weren't flashy, gung-ho murders or Sherlock-esque geniuses [1]. Usually, they were males with weapons experience and familiarity with living on the frontier. Time to track down a target, transport them back, and secure the bounty varied greatly on a case by case basis; it certainly doesn't match pop culture standards. The vigilante narrative is also highly inaccurate. Bounty hunters usually worked for private organizations like Wells Fargo which had an interest in reducing criminal activity and/or were law enforcement officials like sheriffs who needed more money [2] [3] [4].
Believe it or not, bounty hunters did not possess extraordinary powers of prediction. Two factors made the famed skill of tracking a whole lot less relevant. First are the physical constraints. In the area where bounty hunters were employed, the environment was harsh. Take the desolate climate of the Mohave and the dangerous blizzards of the Rockies; running across a river or disappearing into the hills isn't very useful when nature itself seems to want to kill you. Additionally, one must consider criminals' desires. If you had a bounty, you probably had an intense need for infamy and/or wealth. Those can't be acquired if you never contact civilization. Lack of certain resources, environmental harshness, and human desires generally drove criminals to visit towns which made tracking far easier [5].
This leads to the second factor: information sharing. The Wild West equivalent to dusting for fingerprints were aliases and physical descriptions. Bounty hunters were not afraid to ask a lot of questions; sometimes quite forcefully. As the bounty hunter pursued a target, general mandates turned into more specific details. For example, the target was originally said to be within three counties gradually turns into near these two towns and so forth. The expansion of private law enforcement like the Pinkerton's and Rocky Mountain Detective Association aided this spread of information greatly by providing a framework for bounty hunters to cooperate with the incentive being mutual gain. The most poignant example of this information spread "was The Detective, official journal of the police authority and sheriffs of the United States...this five to six page newsletter provided information about wanted criminals: rewards offered, crimes committed, physical descriptions, and when available, pictures [5]." Thanks to information gathering, bounty hunters possessed a significantly lower need for tracking expertise than what is portrayed in movies.
That said, the movies did get on thing right: many bounty hunters possessed elements of expertise and utilized it to their advantage. As mentioned above, those who were bounty hunters had familiarity with guns due to involvement with law enforcement or herding cattle. These traits and the nature of living in a developing West meant many had experience hunting, camping, and so forth which improved odds of successfully navigating the wilderness [5]. In this case, individual examples are the most useful tool to illustrate bounty hunters widely varied skill set. Take the case of Jim Kirker: a former mountain man turned bounty hunter. His experience in navigating mountainous terrain and surviving in harsh conditions aided in killing his targets which were Native Americans [6]. Another example is Charlie Siringo: a former cowboy who became a Pinkerton detective. His work as both a 'classic Western bounty hunter' and a detective who infiltrated gangs undercover demonstrates the shift in tactics due to need for information that I mentioned earlier [7]. In sum, bounty hunters almost always relied more on shared information and gradual narrowing down of suspects and locations rather than Hollywood-style intuition.
Part I - Citations
[1] Man-Hunters of the Old West.
[2] What really happened in the Wild West? The Gunslinger Myth
[3] The History of the American West Gets a Much-Needed Rewrite
[4] Were Bounty Hunters as Despised as Most Westerns Depict?
[5] Rewards, Bounty Hunting, and Criminal Justice in the West: 1865-1900
[6] Did guys actually exist in the Old West who made their living as bounty hunters?
[7] Midnight cowboys: bounty hunters of the Old West
Edit 1: discussion of speed