Why bother tracking TSN after an overhaul?
No, if the part is not subject to a life limit of its root component.

In the airline business, there are actually two terms used in the industry - "overhaul" and "restoration" which can generally be defined as follows:

If a part doesn't have a structural life limit, then the part can be rebuilt to new specifications an indefinite number of times, and each time, it is equivalent to a new part and will generally be called "overhauled" or "zero timed".  Mechanics can track "time since new" but they don't really have to and if parts move around it probably won't be anyway.  Piston aircraft cylinders are like this.  If you buy a rebuilt cylinder, nobody will know how many hours are on it since it was first made.

However a lot of components, especially components that operate under stress on transport aircraft, can have structural life limits applied as a result of endurance tests (where, say, a series of housings under endurance test will start to develop cracks at X thousand cycles or hours, and a life limit will be applied at 1/3rd of the endurance result, which is the safety factor applied to structural cracks that develop in test).

When these parts get rebuilt, they aren't called "overhauled", they are called "restored".  A structural actuator may get a restoration interval applied where it has to be rebuilt (because parts wore out during endurance testing) but once the root component like the basic housing gets to the life limit, the entire part is scrap.  In these cases the rebuild is not an overhaul but a restoration.  The hours or cycles have to be tracked since new since once it hits the life limit it is scrap even if it was rebuilt (restored) the previous month (which an airline won't let happen).