Can Einstein's field equations of GR be expressed without using tensors?

Tensors are required for general relativity. The magic in a tensor is how it transforms with changes in coordinate basis. Tensors give us the tools to describe physics that is invariant under coordinate basis transformations. The reason that we use the notation (and with the Einstein summation convention) is that we can represented invariant physics in generalized notation - which is obviously what we want to do in a representation that doesn’t depend on choice of coordinates.

So look at the convenience of a generalized spacetime metric notation.

ds² = gᵤᵥdxᵘdxᵛ

The above represents ALL metrics that can be obtained from general relativity. The x’s represent any coordinate system, with the u,v indexes representing delineating time and space  coordinates. The expression sums over upper and lower indexes so all 16 metric components are contained in this one expression. For any SPECIFIC coordinate system, you just substitute in the coordinates, e.g. dt, dr, etc. The corresponding metric tensor component will be expressed with t, r, etc. also

The reason that the differential equations relating the components are done with the notation is exactly the same as above. And they obviously can be expressed without the notation - you do that whenever you are solving the equations in a particular coordinate system.

The notation takes a little rime to get used to but it is a very powerful notation for tensors.

You should probably ask tensor questions on the physic forums. Don’t think a lot of math jocks are too familiar ...

Of course. They are 10 nonlinear partial differential equations, with four independent variables. When you expand the Ricci tensor, or the Christoffel symbols, they get VERY long and messy.

You might want to take a look at the Newman-Penrose stuff, if you have good eyes.