Indeed, it's been a pretty fruitful approach. Physicists propose new particles for a variety of reasons. Though they weren't fully recognized as "particles" at the time, gaps in the early periodic table could be thought of as missing particles in a sense.

An apparent consistent failure of conservation laws in a collision experiment could be described as a "particle" if the failure has a quantitative regularity that's consistent with an un-detected particle.

Particles can be proposed to identify a way that a new theory can be tested -- the Higgs and the graviton are in this category. Given a choice, physicists prefer to explore theories that can yield experimental predictions, and particle detection is an area where theories can be tested with decent sensitivity, albeit at some cost.

A very wise person (I wish I knew who) noted:

“All models are wrong. Some models are useful.”