But that’s surely not to be taken literally, right? It’s not that there are literal ball-on-a-spring objects which collectively constitute a field whose oscillations are particles, even if the behavior of these “things” are similar to that of a ball-on-a-spring. What is a field really, then? It surely must be something physical and tangible, not mere Platonic mathematical objects, because particles (and atoms and molecules) are physical and tangible. I don’t think we think about this much in QFT - at some point we just write down an equation (analogous to the plots of oscillations that you drew) and proceed with calculations.
(For this discussion, let’s put aside the caveat that QFT is not a final theory, that it could really be strings, etc. - these only push back the discussion one step back.)