Another supposedly simple question with not so simple answer. First, TCR is far from useless, it is the actual real world variance of resistance with various real world effects on the resistor. There is the 'raw' TCR, this is essentially what you are talking about here, a piece of resistance alloy just laying there, such as a bar sitting on a table top, this bar will have limited effects placed upon it, dimensional variation with temperature is one of them, humidity and barometric pressure (depending on the alloy) also affects this bar sitting on the table. This bar will also have a raw TCR of a specific value which consists of all of the variables affecting the bar in its present state as measured by a resistance bridge. In the case of alloys such as Manganin and Zeranin, the TCR curve is essentially hyperbolic with near zero TCR adjusted around the cardinal temperature point. Other alloys such as Evanohm have much flatter TCRs with wide temperature variations. All TCR measurements, whether it is 'raw' or from a finished resistor is the total net effect of all of the variables that affect the resistive alloy. In the case of wire wound resistors, the 'raw' TCR of the wire is measured and labeled on the spool of wire, this TCR is usually not what you get when the resistor is manufactured, the end result i.e. the final actual TCR of a finished resistor is the net total effects of all of the physical variables on that resistor, in addition to the ones I mentioned earlier, there is additional sources of stress put of the resistive element by the manufacturing processes, for example the act of drawing alloy through a die to produce a specific wire size or the milling of alloy to make film or foil affects the TCR of the element, the 'raw' TCR in effect.
When the alloy is used to make a resistor there are additional mechanical stresses put upon the alloy which further changes the apparent TCR, for wire wound resistors, this is the act of wrapping the wire around a bobbin, in addition to that there is the coefficient of expansion of the bobbin, humidity absorption of the bobbin and encapsulant, ect. all of which affects the overall TCR of the resistor. Of course steps are taken during manufacture to reduce these effects to as low as possible. The end result is a finished TCR which is rarely ever the same as the 'raw' TCR was. The actual TCR of the wire as manufactured can only be changed by very high heat, well above any operating temperatures, in a finished resistor it is physical forces which change the end result TCR, properly made, the TCR itself is very stable long term.
Since Vishay's parts were mentioned I'll give some detail about their manufacture, first Felix Zandman was hardly the only person who developed film/foil resistor technology, he had a lot of help over the years and his initial patents were declared invalid in court. Be that as it may, Vishay became the prime mover of film/foil technology over the years. These resistors are both similar and different than wire wounds, while they both use very similar alloys, the manufacturing processes are considerably different because of the physical differences. Vishay, et al, uses very thin sheets of resistance alloy which are etched by various methods and then attached to a substrate, ceramic being the most common. The act of attaching a very thin sheet of alloy to the ceramic creates considerable physical stresses on the alloy by adhesive, temperature, power, humidity and barometric pressure to name a few. These various stresses created extremely non-linear TCR curves and relatively high TCRs. Over the years, the engineers at Vishay and others, gradually reduced the fluctuations of TCR to ever smaller values, this took a lot of time, decades actually to accomplish. As mentioned there was a lot of 'tweaking' going on with various materials and adhesives trying to find a way to flatten out the TCR curve stresses.
Today they have some models with very low TCR over limited temperatures, while the curve is still not 'flat' per se it is substantially flatter than it used to be. I might add that Vishay does not guarantee their TCRs are 'zero', they guarantee the maximum limit of TCR and I've found over the years that Vishay tends to make claims that they can't always meet. As to Zeranin's comment, "I often wondered how they managed to reasonably cheaply and easily tweak the TCR so ridiculously low on their top-end foil resistors, and that's how!", the process is neither easy nor particularly 'cheap', in fact the yields of these very low TCR resistors I suspect are not particularly high and are likely selected out of a given batch of resistors.
Often, depending on the alloy, the dimensional variance is often one of the smaller variables that affect the overall resistor, Evanohm has a TCE of about 6PPM/°C, the physical variables as I described above have a much more significant effect on the finished resistor and personally I really don't pay much attention to it when I'm making PWW resistors, there are other stresses which are much more important to the finished product and while the dimensional property of the alloy is more important to the film/foil resistor, it also tends to be less of an effect than other stress sources as well.
As resistor manufacturers strive to get to that supposedly golden TCR of zero, all of the various effects become more important and even the tiniest sources of stress become increasingly important, in fact they can actually become more difficult than the larger sources of stress to control as the TCR drops below 1PPM/°C, ask me how I know!
To more directly answer your question in the initial post, you are mostly correct in your premise about dimensional resistivity vs. resistance.
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