What rights do airlines have to deny boarding due to immigration issues? [duplicate]

To give you an idea of what the airline agent is looking at in the airport, here's a Timatic web interface which United Airlines provide:

And once you click on the Check button, you will get:

Note that if you change the "Ticket" field to "Return ticket held", the admission summary changes from "No" to "Conditional".

My question is, to what extent do airlines have the right to reject people from boarding due to immigration issues at the destination country?

They have all the rights. The airplane is their own private property. They don't need any justification to deny you boarding, they can allow or not allow anyone to enter their private property, as they please.

Yes, you did enter into a transportation contract with them, but the fine print in that contract will say something like this.

My logic is that even if I landed in Switzerland without a residency permit and don't manage to acquire the necessary documents then, it should be the responsibility of the Swiss immigrations to kick me out of their country, and not the airline's (which isn't even swiss airlines)?

That is not correct. According to international treaties, it is the responsibility of the airline to ensure that you will be admitted into the country. In case you are denied entry, the airline has to both pay a hefty fine and return you to the airport of origin at their cost. (Although note that the transportation contract with you will also specify that they will recover the fine and this cost from you.)

If you had been denied entry, the airline would have had to pay a fine on the order of 10000$, and fly you back. They then, in turn, would have charged you with 10000$ + the cost of the return ticket, and they would have been completely within their rights to do so.

[…] a recent relaxation of regulation allows me to apply for a residency permit only after landing in Switzerland

Airline gate agents are not international immigration lawyers. They don't have the recent relaxations of all 41412 combinations of origin and destination countries memorized.

They rely on an automated system called Timatic which simply tells them "Yes" or "No". The information in Timatic is maintained based on information from the destination country. So, in this case, Switzerland has put into Timatic that entering without residency permit is not allowed, and that's the only information the gate agent has at their disposal.

I argued with the manager for a while and eventually they let me board after I provided enough evidence that I had sufficient reason to be living in Switzerland and as a result I almost missed my flight.

You were lucky.

Moral of the story: Don't rely on gate agents being aware of every recent change in every immigration law of every country. Make life easy for them.

Even if you are right, arguing the fact that you are right may almost certainly take longer than the airplane is going to wait for you.