### Solution:

#### Part (1):

Let's denote the capital of the live streaming platform at the end of year $n$ as $a_{n}$. The process to calculate the capital at the end of each year after deducting operating costs and reinvesting the remaining funds can be described as follows:

- For the first year:
$$a_{1} = 500 \times 1.4 - 100 = 700 - 100 = 600$$
- For the second year, using the capital at the end of the first year:
$$a_{2} = 600 \times 1.4 - 100 = 840 - 100 = 740$$
- For the third year, using the capital at the end of the second year:
$$a_{3} = 740 \times 1.4 - 100 = 1036 - 100 = 936$$

Therefore, the capital of the live streaming platform at the end of the third year after deducting operating costs is $\boxed{936}$ million yuan.

#### Part (2):

To find the maximum annual operating cost $a$ that allows the platform to reach 3000 million yuan at the end of the sixth year, we use the formula for the capital at the end of year $n$, considering the deduction of operating costs each year:

- The formula for the capital at the end of the sixth year, $a_{6}$, considering the geometric series for the deduction of operating costs is:
$$a_{6} = 500 \times 1.4^{6} - a \cdot \left(\frac{1-1.4^{6}}{1-1.4}\right)$$

Given that $a_{6}$ must be at least 3000 million yuan, we set up the inequality and solve for $a$:
$$500 \times 1.4^{6} - a \cdot \left(\frac{1-1.4^{6}}{1-1.4}\right) \geqslant 3000$$

Solving this inequality for $a$, we find that:
$$a \leqslant 46.8$$

Therefore, the maximum annual operating cost should be controlled at $\boxed{46.8}$ million yuan at most, in order for the live streaming platform to reach 3000 million yuan at the end of the sixth year after deducting operating costs.