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No, the analogy is more that the oscillations are themselves the particles.
The addition of energy into a system would be this hand push. The fact that the particles themselves exist means that they are oscillations in this mesh (with some energy/frequency). Interactions with other particles can add or remove energy.
Definitely these canvas metaphor are just conveniences. Also, I got it from Zee’s “Quantum Field Theory in a Nutshell” which is a standard graduate or advanced undergrad level book on QFT.
Do you have an easily digestible source so I can stick this in the face of conservatives with classic talking points?