[1] - https://en.wikipedia.org/wiki/Hashcash
[2] - http://hashcash.org/papers/pvp.pdf
[3] - https://moderncrypto.org/mail-archive/messaging/2014/000782....
The anti-spam system is basically what you said: you spend a small amount of bitcoin, and the server grants you some large number of "tokens". These tokens can be used to perform actions which use resources on the server, either storing messages, or getting messages. The exact price is configurable, and currently set low [1].
There is a second mechanism that I use, which is intended to lower costs for the average user, but still prevent spam. Essentially, in each bitcoin transaction, a small "transaction fee" is required, which goes to the miner. It is possible to prove, via a cryptographic signature, that you were the originator of a transaction, and therefore the person who spent that transaction fee. The bitnet system grants tokens to people who can prove they spent money on transaction fees [2]. The idea being, a typical bitcoin user will accumulate transaction fees anyways, but a spammer will have to go out of his way to send bulk messages.
[1] https://github.com/ortutay/bitnet/blob/master/bitnet.go#L50 [2] https://github.com/ortutay/bitnet/blob/master/bitnet.go#L38
The idea seems to be that you provably "burn" a small amount of Bitcoin to get an identity. An innocent person can then carry on using that identity forever without doing any more computation. Meanwhile a spammer will ruin that identity's reputation almost immediately and then have to pay again to get another one.
There's also botnets, whose resources hijackers are happy to exploit.
Interesting that SMS appears to have simply legislated the problem from existence (or so says the article), I suppose if anyone can connect to a telephony network for cheap the problem would persist there too.
OpenBazzar a bit-torrent like p2p version of ebay uses proof of burn to make bad behavior expensive.
If all messages are seen by my phone, just to be inspected and discarded, then my phone is consuming bandwidth and processing cycles on irrelevant data. And unless my phone has free bandwidth, and the processing cycles don't consume power (or it has infinite power), how can it ever not be a factor?