I used to think so, but these days, I wonder why I used to think that rest mass is more intuitive. For most things we interact with and develop intuitions about, their speeds are so low that relativistic and rest masses are virtually the same, so that intuition can't come from experience. Photons are the exception to that, but we never encounter them at rest, and there's no particular reason we would have an intuition that photons must be massless until we learn about it in school. And then people have to overcome the confusion that photons do still carry momentum despite being massless.

If we think of mass in terms of inertia, the difference between rest mass and relativistic mass only shows up at relativistic speeds (like in a particle accelerator, or cosmic rays). At that point, relativity and Lorentz transforms needs to be taken into account anyway; the half-lives of unstable particles get affected by time dilation, and so on.

The other way that mass shows up in physics is through gravitation. If we think of mass in terms of its gravitational influence, then relativistic mass is all that matters, and rest mass seems to mess with people's intuitions much more, causing all sorts of conceptual mistakes. Like thinking that photons don't gravitate.

Rest mass does has some theory-simplifying mathematical advantages, but I don't think those benefits will matter to the intuitions of non-experts.