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

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.

Airlines, in general, have an absolute right to deny boarding. But in the case of immigration issues, they frequently have an obligation to check whether passengers can be admitted to the destination. Many countries will levy fines on airlines who deliver a passenger to their border who is lacking required documents.

Every country implements this differently, but it is common practice that airlines must either take responsibility for removing the inadmissable passenger, at their own expense, or pay a fee or fine to the government if they’re unable to do so.

In Switzerland, the law is as follows:

Air carriers transporting persons must take all reasonable measures to ensure that they only transport persons who possess the required travel documents, visas and residence documents to enter the Schengen area or to travel through international transit zones of the airports.

1. The air carrier is obliged at the request of the competent federal or cantonal authorities to provide immediate assistance to any passengers that it is carrying who are denied entry to the Schengen area.
2. The obligation to provide assistance covers:
   1. the immediate transport of the person concerned from Switzerland to their country of origin, to the state issuing the travel documents or to another state where their admission is guaranteed;
   2. the uncovered costs of the required attendance as well as the customary subsistence and care costs until departure from or entry into Switzerland.
3. If the air carrier is unable to provide evidence that it has fulfilled its duty of care, it must additionally bear
   1. the uncovered subsistence and care costs that have been covered by the Confederation or the canton for a period of stay of up to six months, including the costs for detention under the law on foreign nationals;
   2. the attendance costs;
   3. the deportation costs.