Cost at Each Level of a Recursion Tree
You are right!

The work $W$ at each node (of level $i$), splits into $Wa_1, Wa_2, \dots, Wa_k$ at level $i+1$.

If you add them up, you get that the total work at level $i+1$ is $a_1 + a_2 + \dots + a_k$ times the total work at level $i$.

Of course, this assumes that your $\Theta(n)$ is some exact $cn$, but this argument is good enough to prove that $T(n) = O(n)$. Proving $T(n) = \Omega(n)$ is trivial.

btw, I recommend you look at Akra-Bazzi as a useful tool, if you are interested in such questions.