Is third-party purchased liquidity also possible using liquidity advertisements and dual funded channels?
It is not currently supported, but Lisa Neigut (niftynei), the primary developer of both dual funded channels and liquidity advertisements, seems to believe it's possible. Two approaches were suggested:
Onchain PSBT workflow
Bob publishes a liquidity advertisement (currently supported)
Alice notifies Bob that she wants to fund a channel for Carol. This is not currently supported by the protocol and would need to be arranged out of band.
Carol requests a channel with liquidity from Bob (currently supported)
Instead of asking Carol to fund the channel open and pay for the liquidity (currently supported), Bob asks Alice (not currently supported)
Alice sends a PSBT with funding to Bob, Bob returns his half of the PSBT, they both sign and broadcast the funding transaction (like the current protocol but with Alice substituting for Carol).
After the funding transactions receives enough confirmations, the channel is fully opened to Carol (currently supported)
Analysis: the security of the above is equivalent to the long-term security of Lightning Labs Pool---onchain protection against double spending. Users who trust Pool not to double spend can opt-in to a short-term weaker security model that allows them to begin spending sooner; that weaker security model and its speed can't be duplicated for liquidity advertisements without additional work.
Offchain workflow
Bob publishes a liquidity advertisement (currently supported)
Carol requests an offchain invoice for some liquidity from Bob; Bob sends the invoice (sending invoices is currently supported but you'd need a plugin or other software to actually generate these invoices and perform the correct actions when they were paid)
Carol forwards the invoice to Alice, who pays it and sends Carol the proof of payment (giving invoice details to another node and them paying it is currently possible; transferring a proof of payment is also currently possible; again, you'd need software, or manual human intervention, to actually do all of that)
Carol includes the proof of payment in a request to Bob to receive a channel with liquidity. (not currently supported)
Bob opens the channel. (currently supported)
Analysis: this method requires that Alice trust Bob. It may be possible to reduce some of that trust by using an HTLC, but that isn't perfect and adds complications related to hold invoices if it takes the channel a long time to get adequately confirmed onchain. However, the amount of money Alice pays Bob upfront can be as small as the amount of transaction fee needed to confirm the channel onchain---Alice can wait to pay the actual lease cost until after the channel to Carol is fully open. This also means that leases don't need to have a fixed duration and that Alice may even vary subsequent payments to Bob based on Carol's reports about his performance as a routing node.
Based on the above, I think third-party purchased liquidity ("sidecar channels") is clearly possible to implement with LN liquidity advertisements and dual-funded channels, although it is not available
now.
Please note, all of the above is my own interpretation of a technical discussion about experimental new technology. It may contain errors, and those would be my fault, not Neigut's or anyone else's.