However, this will take at least 10 minutes to be confirmed, and you would have to do it with every party you wanted to transact with for the first time.

Not necessarily. The way that some of the LN clients currently work is that they will automatically establish several channels with other nodes for you when you first start the client. So when you want to pay for something at a merchant that you do not have a channel with, you can simply route your payment through one of your open payment channels. There is no need to create a new channel with each individual person you wish to transact with; that's the whole point of being able to route payments.

So ideally, wouldn't there be some sort of single, intermediary service or protocol that I could enter this multi-sig wallet with instead?

Ideally, no. That would mean that there is centralization. What would actually happen is that both you and Starbucks have open payment channels with other people, and you simply route your payment through another person.

While I assume there is some other answer to this, here is my understanding of it:

One of the key features of LN is that Channels can stay open indefinitely:

as long as the two parties in the channel continue to cooperate with each other, the channel can stay open indefinitely -- there is no mandatory timeout period.

This means that you can open a channel with Starbucks and reuse this channel for all your future transactions. Of course the opening of the channel (the funding transaction) and closing it has to be on-chain but as with many things in life you have to look at this as an investment - you pay/wait now and over time you get your investment back and even profit from this.