I found many research papers on differences between Teflon and bronze die for pasta extrusion but there are no comparisons between bronze vs steel/brass/any other metal dies. Why we still stick to bronze dies instead of using steel as cheaper alternative? Are there any differences of pasta structure after extrusion (steel vs bronze)? 
I too have been trying to find a satisfactory answer to this question, as it seemed like bronze was wheeled out for primarily marketing reasons and served more of a default when not using PTFE, rather than being honestly motivated by material properties. This, combined with some food pseudoscience, can be pretty frustrating. My reading of the situation has been that Teflon/PTFE is effectively superior in every way for dried pasta producers. It presents minimal extrusion friction and heat, is easy to machine and replace, creates pasta that looks good on shelves, and even improves shelf life of dried pasta due by reducing likelihood of grain weevil infestation, leaving the no-PTFE producers in a high-cost niche that feeds on marketing. However, in more recent reading, it seems like bronze may be more of an intentional choice. I do not have a hard answer to your question, but the literature seems to circle around a few key aspects, particularly heat dissipation. I hope that, even if this doesn't answer the question, I can give you a few sources and bibs to continue the search. Some Reading Extruding and Drying of Pasta, Manthey/Twombly; technical overview of extruded pasta Pasta. History, technologies and secrets of Italian tradition, Barilla; lighter read, but gives a lot of context The Study of the Behaviour..., Trasca; not enlightening but a good bibliography and gives some context around motivations for PTFE that can be used to backsolve a little Example Barilla die insert from Barilla Archives Takeaways from Reading Twombly, Pg 8 Sec 6 "Die support is made from bronze or stainless steel. The support must be capable of withstanding a tremendous amount of pressure over time without yielding. Stainless steel can tolerate higher pressures, but tends to retain more heat than bronze" "Inserts are generally made from bronze, due to their low heat retention." This is what I see again and again. Bronze is chosen for its thermal properties, and it's specifically chosen over SUS even when other parts of the die assembly are made from SUS. You've noted above that the thermal conductivity of the two metals are not incredibly different, but bronze's k will depend a lot on the copper content, and it's not rare to find >70W/m-K bronze. SUS, in contrast, tends to have quite low (<25W/m-K) thermal conductivity. For high-volume pasta production, this would result in a large difference in pasta extruded over time, as SUS die inserts would need to be run slower. I have not been able to find the specific alloys used in the insert manufacturing, but would assume from the surrounding data that the bronze is a high-copper alloy. Twombly, Pg 7 Sec 4 "Excess heat generated by friction during extrusion is removed by use of a water jacket which surrounds the extrusion barrel. A high volume of warm water is used to maintain both the barrel and dough temperature near 45°C." Worth noting. Keep in mind that a water jacket cannot be used (without complex cooling channels) for the aperture end of the extruder due to the fact we're extruding dough. So, you want to move heat energy away from the insert contact surface as quickly as possible so that it can be carried away by the water cooling. Barilla Archives, Picture 2 Depiction showing that the largest pasta producer's PTFE dies are made from bronze and PTFE, not steel and PTFE. This suggests that there are still properties of the bronze that are still desirable, even when the dough does not contact the metal, the company cannot advertise using a bronze die insert, and when the producer has more than enough capital to afford machined stainless steel. This backs up the thermal properties theory. It may also just be the case that the economics of SUS don't work out here. The lifetime of the insert will be higher, but the machine will need to extrude slower, in order to account for thermal conductivity, and the production costs for each insert will be higher. Barilla, Pg 111-112 Apparently, Barilla's pasta die supports have been cut from what Barilla calls "Bral," their name for an aluminum-bronze alloy. I have not seen this mentioned in other publications, but makes sense if they are trying to balance thermal conductivity and pressure tolerance. My understanding is that aluminum-bronzes should beat out bronze in practically every aspect that matters here, so that reason may be significant to the reason that bronze is chosen. I have no experience buying nor machining that alloy, but it seems like it may be more expensive and harder to machine than bronze. Barilla, Pg 65 (Bonus History) "Dies were made with materials such as copper, red bronze, manganese bronze, etc., which were resistant to the acids formed during the fermentation process." I'm going to guess that the origin of bronze as the material of choice all comes from the high workability of the metal combined with its chemical properties. At the time (~17th century), this was probably the best that metallurgy could do (more research needed). It's interesting to consider that the thermal characteristics would have basically been mostly irrelevant until the dies were worked much harder. 