Is it reasonable for clients to ask for indemnifying by service provider unlimited liability towards any lapse or mistake during performance of the services

As a client, we see it the other way where the service provider contracts will state that the provider has very limited liability which we see as very unreasonable.

It's a great concept, but one that you will have a very limited ability to accomplish, especially when dealing with large companies. (It is almost impossible to accomplish any form of modification to their agreement/contract language.) I am also uncomfortable with the all or nothing approach. In real life, we should be responsible for the things we have control of, as well as practice good 'defensive driving' for the others out there with us.

In the world of live theatre, I did have a great deal of luck working with my partners (the show producers traveling and performing in my venues). I developed language that basically said we each agreed to indemnify each other to the extent we were responsible. I found almost no push back when I walked them through it.

To explain it here: there are generally two areas of operation in a theatre - the stage and behind/below for the producers and cast, and the auditorium and in front for the building operators and the audience members. Indemnification language used to read that anything that happened in the auditorium or lobby would be 100% the responsibility of the building operator until shows like The Lion King arrived. As soon as the cast was in my lobby and auditorium mixing it up with the audience I was no longer interested in offering 100% coverage for something they might do. At the same time, if one of my crew members did something on the stage or in a dressing area resulting in harm or injury, the show didn't want to cover me either.

In the world of technology, I don't think you will have the same cooperation and requesting they cover unlimited liability won't be accepted anywhere. You do have the right to get their disaster recovery plans and then plan your own accordingly. Make sure to have your insurance company involved as well. Whether the businesses or lawyers like it won't mean much when the insurance company declines to accept language upon receipt of a claim.