I'm not sure if the wheel is unbalanced, or the tyre is damaged.
Going faster I don't feel the shake that much, the shaking is less pronounced, has it's slowing down, the shaking becomes stronger and more pronounced. If it's unbalanced the same shaking should occur at high speeds with the same intensity or worst.
It could be the tyre, I'm not questioning that, and I do believe is what makes more sense.
But also wonder if it could be something wrong the the direction bar, rim, suspension ....