I, too, say that massless objects do not exist in Newtonian mechanics.
While it is true that first-year texts abound with examples that use "massless" strings/springs, it is also true that these same texts point out that out that the strings/springs are not massless. A random sample of three popular first-year texts (itailcs and bold below are used in the texts):
Halliday and Resnick "In the special case in which the weight of the spring is negligible ..."
Serway and Jewett "In problem statements, the synonymous terms light and of negligible mass are used to indicate that a mass is to be ignored when you work the problems."
Knight: "Often in physics and engineering problems the mass of the string or rope is much less than the masses of the objects that it connects. In such cases, we can adopt the massless string approximation. In the limit ##m_s \rightarrow 0##, Equation 7.8 becomes ..."
When I teach first-year physics, I use the term "negligible" before taking ##m=0## in the first few such examples that I present.