The communication from satellite to satellite using laser links is
much faster than that using fiber cables on the ground.
This is pretty much the opposite of the truth. Laser in the fiber is always more stable. Had this been the truth, then the global communication wouldn't predominantly rely on fibers now. But this is irrelevant because to the user, the bandwidth is limited to the mmW link between the satellite and the user.
The speed of light within the glass fiber is only about 66 % of the
vacuum speed of light valid for the lasers in the Starlink orbits.
This is relevant to delay, but not to bandwidth. (The highest bandwidth data transport currently is to physically ship hard drives using airplanes. Delay is terrible but bandwidth is incredibly high.)
A Starlink network with users only does not make sense
I don't totally disagree with this statement because I don't think the current proposed user base and business model makes a good business case either.
so what about the servers within that network?
Maybe, but the use case seems narrow.
First you need to have a server farm in a location where no other communication channel is cheaper. This is possibly be deserts or polar areas where electricity or cooling is much cheaper but not accessible via fiber. Pickling the geo location for a data center is always a tricky trade-off between the cost of cooling, electricity and communication but I suspect the true savings on cooling and electricity when you go to the extremes of no ground commutations at all.
At the same time you don't want to be near any other users because the servers could easily saturate a satellite and makes it unusable to other users. Remember each satellite only have 17Gbps bandwidth and modern servers are routinely equipped with a few 10Gbps or even 40Gbps links. So it gets expensive quickly once you scale up the server farms and need more exclusive satellites.
One possibly viable use case could be CDN where the network traffic goes the opposite direction of end users, e.g. the uploader side of video streaming. CDN servers are always built physically close to end users so it won't be a bad idea to place scatter a few CND servers across the globe when the end users are also scattered.