First of all, maybe Harry just ignores these "scientific" methods. However, the participants had ample time to prepare for the final task, and I find it difficult to believe that Hermione did not know of these strategies:
Finally, in the last week of May, Professor McGonagall held him back in Transfiguration.
"You are to go down to the Quidditch pitch tonight at nine o'clock, Potter," she told him. "Mr Bagman will be there to tell the champions about the third task." [...]
"That's right!" said Bagman. "A maze. The third task's really very straightforward. The Triwizard Cup will be placed in the centre of the maze. The first champion to touch it will receive full marks." [...]
"There will be ostacles," said Bagman happily, bouncing on the balls of this feet. "Hagrid is providing a number of creatures ... then there will be spells that must be broken ... al that sort of thing, you know. [...]"
[Chapter 28 - The Madness of Mr Crouch]
Harry's nerves mounted as June the 24th drew closer, but they were not as bad as those he had had before the first and second tasks. For one thing, he was confident that, this time, he had done everything in his power to prepare for the task.
[Chapter 31 - The Third Task]
Then, the right-hand strategy assures that you either find the exit or return to the start, which is surely not a desirable outcome, especially as there are other contestants.
In this example using a right-hand or left-hand approach always gets you back to the entrance (and I guess the task maze would be more complicated!)
Even supposing we are in a situation where the right-hand rule would make you find the exit, the path could be extremely long and convoluted, again a disadvantage in a competition; a longer route, besides taking more time, also means a higher chance of bad encounters along the way.
In this example the cup is eventually reached using a right-hand approach, but only after having explored the dead end on the right side; turning left and then right makes for a much shorter path.
And I think this is the main argument against using a fixed strategy: the magic encounters. The right-hand rule can work only if you are always free to choose your direction at each fork, which is not the case when magical creatures are involved! When Harry meets a Skrewt he is forced to make a U-turn:
Then he rounded another corner, and found himself facing a Blast-Ended Skrewt. [...]
The Skrewt was inches from him when it froze - he had managed to hit it on its fleshy, shel-less underside. Panting, Harry pushed himself away from it and ran, hard, in the opposite direction [...].
And again, upon meeting the sphinx Harry is given a chance to pass her (which he is told it's the quickest way) only if he guesses the riddle:
If the riddle was too hard, he could keep silent, go away from her unharmed, and try and find an alternative route to the centre.
If you cannot always apply the right-hand rule, there is no point in applying it at all.
Finally, as user @damien-lavizzo already said, a scientific method could not be the right choice when magic is involved. However, re-reading Chapter 31 - The Third Task I could not find references to the paths shifting and changing, so that could just be a movie thing.