Intuition behind the coupon collector problem. Is there inclusion-exclusion principle in play?

Yes, it is exactly inclusion exclusion. For each $t\in \mathbb N_0$ and $k\in [n]$, let

• $E_t$ be the event that not all the coupons have been collected after $t$ coupons have been drawn,
• $E_{k,t}$ be the event that coupon $k$ does not appear in the first $t$ coupons.

Then $E_t=\bigcup_{k=1}^n E_{k,t}$, so inclusion-exclusion applies. It all comes together with these three pieces: $$ E[N]=\sum_{t\ge 0}P(E_t),\\ P(E_{t})=\sum_{S\subseteq [n]} (-1)^{|S|+1}P\left(\bigcap_{k\in S} E_{k,t}\right),\\ P\left(\bigcap_{k\in S} E_{k,t}\right)=\left(1-\sum_{k\in S}p_k\right)^t. $$ To be clear, $S$ ranges over only nonempty subsets of $[n]$.