Yhyjyk

The story is about an alien race that comes to Earth and takes a single Earthling to help man a ship to fight an invasion from another people. The human is a psychic as is everyone else on the ship. The main fleet is made up of the alien race, but the ship the human is sent to is crewed by single psychics from different planets. The human has to fight his way up the chain to eventually command the ship. He refuses to retreat when ordered to do so and leads an attack on the invading aliens.

This sounds a lot like Gordon R. Dickson's 1970 novel Hour of the Horde, which was previously asked about and answered here.

The protagonist is selected by aliens as Earth's sole representative in an intergalactic war. He does have a kind of sensitivity that is not typical, and which is essential to operate the weapons to be used in an upcoming battle to divert an enemy fleet from our galaxy.

After his selection, the protagonist is placed on a very small fighting ship in the dregs of the defense fleet, where he is basically told to stay out of the way. Each crew member of the ship is also a single representative from their planet:

The room was full. On its furniture and around its walls, stood and
sat a variety of different-appearing beings. All were four-limbed,
standing upright on the lower two and with hand-like appendages at the
end of their upper pair. They were all roughly the same size and
proportion and general shape. But there was tremendous variety.

No two of them had the same skin color. No two of them had the same
facial appearance. All had roughly similar features, as far as
possessing two eyes and a single nose and a mouth was concerned. But
from there on everything was different. Their appearance ranged from
that of the completely innocuous, to the completely ferocious — from
one being who seemed as round and inoffensive as a toy bear to one who
seemed a walking tiger, equipped with a pair of ripping teeth
projecting over his lower lip from the upper jaw.

Dissatisfied with these instructions, and unhappy with the acceptance of this by the other aliens in his ship's crew, he battles each one on one in order to rise through the ship's informal command hierarchy.

Eventually, he secures the cooperation of everyone else on the ship. When the climactic battle comes, the commanding aliens order a general retreat, which he ignores and instead charges the enemy fleet alone:

“Retreating?” echoed Miles. "Retreating — you mean just we little
ships are retreating? Or more than just us?”

"Haven’t you been informed?” roared the harsh voice, above him. “The
Center’s computational devices have calculated and found an answer
that predicts defeat if we try to stop the Horde. All are leaving. All
— ”

The voice was cut off suddenly, as Miles jabbed at both voice and
sight communication controls. Abruptly, in the screen before them,
formed a schematic of the whole Battle Line. It showed the whole Line
from end to end, and the ships in all their sizes and varieties, but
as if only a few yards separated them. As Miles, Luhon and Eff
watched, ships were winking out of existence in that Line. Even the
huge globular Dreadnaughts of the Center Aliens were disappearing.

It was true. After everything — after all their work and the work of
the Center Aliens and others to set up this Battle Line — now just
because of some cold answer given by an unliving device, the greatest
strength the galaxy could gather was not going to face the Horde after
all. They were all going to turn tail and run, save themselves, and
let the Horde in to feed upon the helpless worlds they had been sent
out here to protect.

...

Before Miles, in that moment, there also rose up a picture of his
people and his world — the world as he had seen it, during those last
days when he had moved like a ghost from spot to spot about its
surface, and among its many people. He saw it, and at the same time in
his mind’s eye, he saw the picture of the world that the two Center
Aliens had shown him — the world that a million years before had been
cleaned to the point of barrenness by the Horde.

In his mind’s eye now, he saw Earth like that. One endless,
horizon-wide stretch of naked earth and soil, with nothing left.
Everything gone — all gone. The cities, the people within them, their
history, their music, their paintings, Marie Bourtel...

“I won’t!"

...

Miles hands slapped down on the console in front of him. To his right,
Luhon’s flashing gray fingers were already blurring over his controls,
and Eff was busy at his left.

Like a living creature with one mind, the Fighting Rowboat lifted
from its cradle and flashed into shift — single-handedly and alone
into attack against the uncountable numbers of the Silver Horde.

This throws off the enemy fleet's configuration enough that the defensive commander sees an opportunity to attack and reverses the retreat, subsequently winning the battle.

Per ISFDB, the story was originally published in the May 1969 issue of Venture Science Fiction Magazine, and it can be read online in the context of its original publication courtesy of archive.org.