Realistically, mandatory serial numbers are a means of government control, so that ordinary people can be forbidden from having legal guns.
One of the ideas, is if a gun is used in a crime, and the gun is traced back to the legal owner (mandatory gun registration), then the gun owner will be found criminally responsible.
The idea is to keep making gun ownership more onerous and legally dangerous, so as to render gun owners a small, politically ineffective minority.
Maybe I have should have included a sarcasm tag.....
Federal regulations related to serial numbering have also been used against firearms manufacturers. A few years back (IIRC), one of the major AR makers got into 'hot water' with Uncle Sam, for producing lower receivers that lacked serial numbers. The receivers were held in reserve by the manufacturer, to replace other receivers that might be returned for replacement under warranty. For example, if AR lower with serial number S12345 was returned by the owner, and could not be repaired, then (1) the defective unit would be destroyed, (2) the old serial number would be applied to a 'blank' lower, and (3) the new lower with SN S12345 would be shipped to the owner. For some reason I do not recall, the feds took exception to that process.
On the plus side, the application of serial numbers to firearms can have benefits, in terms of identifying safety issues, design changes, etc. Early Springfield 1903 rifle receivers were apparently subject to failure under certain conditions; Rock Island Arsenal improved the heat treatment process at SN 285,507, and changed to an improved steel alloy at SN 319,921 (which is worth knowing, if you own an RIA 1903). Ruger made some major design changes to their 9mm P-series pistols, beginning at SN 304-70000, which you should know before buying magazines (or trying to swap to a Ruger factory .30 Luger barrel - ask me how I know ;^).
Serial numbers (when they exist) may be of interest to collectors, as well. A gun owner down in Texas had inherited his dad's Winchester Model 70 (which had been purchased from the 'Coast to Coast' hardware store in Durango, Colorado), and used it for years as his 'truck gun'. At some point he looked at the serial number, and realized that he owned the very first Model 70 ever produced (Serial Number 1), which was apparently worth in excess of $100,000...