Is it possible to watch the chain to count the number of Lightning channels?
They can only be detected when force closed and there are still pending htlcs otherwise they look like a spend from a 2-2 multisig wallet. I've been told some people try to do chain analysis and machine learning to predict private channels. I highly doubt that this is reasonably possible.

Although there is one thing to consider. If a node has public and private channels the funds get mixed. Implementations currently do not do separate bookkeeping of those outputs which is a privacy issue and an attack vector for what you want. However if a new ln node is set up and it only has private channels it will be hard to detect those on chain (unless the Chanel partners later reuses / mixes the outputs with some fire public channels.