It is not a width, it is a depth you are referring to. The rack rails need a rack that has both front and rear ear rails, that are a minimum of 570 mm apart.
The problem is if the cabinet is not deep enough you are likely to run into other problems like the chassis itself will not fit into place with enough clearance front and rear for cables etc.
You have completely misunderstood the different measurements and what they mean.
A standard rack is specced as 19" or 482 mm. This refers to the maximum width of a chassis that will fit into such a rack. That Silverstone chassis is a 19" width chassis. It will likely fit inside your rack no problem.
The measurement for the Silverstone rails is likely the minimum depth, not width. These rails are designed to attach to the vertical rails that are at the front and rear of a rack. To properly support a large bulky chassis like this one the sliding rails are attached to both the front and rear vertical rails in the rack.
All you need to ensure is your rack has at least 570 mm distance between the front and rear vertical rails to attach the Silverstone sliding rack rails for your chassis.
My primary system is 10 years so I'm looking at the same kind of question right now. The thing is, anything you buy today will bea huge leap in performance forwards regardless. There is not that much gain between two generations and it depends vastly on application or OS capabilities what the actual performance gains are.
For example I just upgraded a 2 year old system from Pentium Gold to i5 with the same motherboard. There was a noticeable performance gain with one applications that is multihtreaded being 10x faster in one aspect of it, but other less optimised operations are only a little faster. Two generations of i5 you wouldn't see such a leapfrog of performance.
I buy the open racks that have adjustable depth e.g.
https://www.startech.com/en-us/server-management/4postrack12u
these are a range of different heights and the depth can be adjusted from 56 to 1020 mm.