One of the main reasons the virtual particles are used is that in many contexts we do not have a non-perturbative formulation of quantum field theory. What we can do is compute some amplitudes perturbatively (e.g. for outcomes of particle collisions) using Feynman diagrams. These diagrams have input/output lines in them, usually identified with colliding particles and collision products, but also intermediate lines that start at one vertex and end at another, staying entirely within the diagram. By extension, these came to be interpreted as virtual particles. One can also imagine diagrams without input/output lines at all, which would correspond to creation/annihilation of virtual particles in the vacuum.
So virtual particles are at least useful in the way pictures of chemical bonds are useful in chemical calculations, although from quantum-mechanical point of view these "bonds" are ephemeral. One place where virtual particles had heuristic value, for right reasons or not, is the prediction of Hawking radiation based on semi-classical mixing of QFT and general relativity near the horizon of a black hole. Out of a created virtual pair one particle falls below the horizon, and the other acquires escape velocity producing the radiation. This picture is suggestive, Hawking himself suggested it in Breakdown of Predictability in Gravitational Collapse (1976), and perhaps it served as his motivation, even if now there are ways to derive it without virtual particles. Here is Parentani's 2010 paper From Vacuum Fluctuations across an Event Horizon to Long Distance Correlations that utilizes Hawking's picture. The same heuristic value attaches to other "appearances" of virtual particles.
Are the virtual particles "real"? At this point we are not even 100% sure that there is something non-perturbative that QFT approximations approximate, let alone if a computational tool for these approximations can be projected onto reality. Some of those who believe that non-perturbative QFT exists expect that it will not be interpretable in terms of particles at all, or fields for that matter, but that undermines far more than virtual particles, see Baker's Against Field Interpretations of Quantum Field Theory. Even interpretations of quantum mechanics, where we do have a mathematically impeccable non-perturbative formulation, are still controversial. However, ideas about atoms and electrons in the 19th century were mostly wrong from modern point of view, but nonetheless provided valuable heuristics for developing modern theories. Ideas about ether helped Maxwell formulate his equations, even if later its existence was rejected. Virtual particles may end up in the same basket.
