The German physicist Rudolf Haag presented a new approach to QFT that centralizes the role of an algebra of observables in his book "Local Quantum Physics". The mathematical objects known as operator algebras (C* and W*) seem to have a lot of importance from this perspective.
I have heard opinions for and against algebraic quantum field theory. A common argument for AQFT that I hear often is that it places QFT in a formal mathematical universe, but I don't really know what that even means. There are other approaches to quantum mechanics (using techniques from microlocal analysis, for example) that emphasize rigor, but do not necessarily deal with an algebra of observables.
I have heard more opinions rather than persuasive arguments against AQFT. The one argument that I am aware of is that AQFT does not deal very well with the Standard Model, which is one of the big empirical success of particle physics and plays a central role in mainstream (Lagrangian) Quantum Field Theory.
I am wondering how AQFT's treatment of the Standard Model (among other things) creates a problem for physicists.