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Look at it this way, if x = number of total users, routing nodes will grow O(log x), not O(c^x). The users can grow exponentially, the routing nodes won't because the marginal cost of processing an extra transaction from an end-user is very close to 0.

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I thought the routing node was putting up some amount of Bitcoin per channel. Each channel would therefore have a non-zero cost (and each user requires a channel).

I would need to go back and refresh my understand as it's been quite a while since I read the LN whitepaper, much less kept abreast of developments in that space.

When you fund a channel, the Bitcoin is still yours, you're not giving it away.

With batched channel open/closes, it'll be super cheap to open and close channels. The only cost you'd pay is the opportunity cost if your counterparty isn't sending transactions, so you're not collecting routing fees. If that's the case, you can just close the channel if you need to fund another channel and don't have spare Bitcoin for it, but that's about it.

You've just outlined how the marginal cost that you originaly stated was near zero, is in fact quite a lot larger than zero (you can't open channels without locking up your node's Bitcoin). Also, each channel opening costs a Bitcoin L1 transaction fee (which is a lot larger than $0.01 and can only really grow from here).

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