Why does the Darksaber get heavier for Mando in The Book of Boba Fett?
That's because of the Kyber crystal in it.
In Disney/Lucasfilm canon, light side users of the Force don't have that problem because they are the ones that awaken the Kyber crystal.
But Sith do need to bleed their crystals.
Each Kyber crystal is alive and has some kind of will of its own. Light side users have no problem using them, but for whatever reason dark siders need to break the will of their Kyber crystals.
And that reason seems half revealed with the Darksaber. Where if they don't bleed their crystals then what was happening to Din Djarin, would happen to the Sith. But not because of something as simple as mental state, rather Kyber crystals don't react well to the dark side users.
Unlike what some like to say the Kyber crystal is not the power source of the lightsaber. The Kyber crystal just focuses the energy of the real power source of the lightsaber.
“It's a Kyber crystal, which focuses the power of the lightsaber beam.”
- Star Wars: Master&Apprentice -
When Sabine Wren was training to use the Darksaber, Kanan Jarrus explained to her why the Darksaber was getting heavier.
He said
“Energy constantly flows through the crystal. You're not fighting with a simple blade as much as you are directing a current of power. Your thoughts, your actions, they become energy. They flow through the crystal as well, and become a part of the blade.”
So because the Kyber crystal in the Darksaber is alive and it's the crystal that focuses the energy of the beam, then the mental state of the user influences the energy in the blade.
Din Djarin was having difficulty to use the Darksaber because his mind wasn't focused, and as the saber became heavier he thought he could compensate with physical strength. When the solution is mental stability.
Many people have used the Darksaber and they didn't need training to use it.
In The Clone Wars, Pre Vizsla while fighting Obi-Wan with the Darksaber he had no problem with the weight of the blade.
The problem was in Din Djarin and his mind, not in the Darksaber.
Those at Lucasfilm unknowingly gave a good explanation as to why the Sith need to bleed their Kyber crystals, and they didn't even realized it.