The first thing I would do is ask the question: how good are they?
The comments to your question have already hinted at the answer. Not so good.
All-to-all connectivity is a dream for some platforms (e.g. superconducting) and very hard to realise on others (e.g. trapped ions). In fact, I would be happy to have an all-to-one to start with, given that it is a good CNOT connection. Note that the error rates of CNOT gates are much higher than Rz rotations.
With a thousand qubits and say a constant depth of five, I would try a very simple problem of finding the ground state energy of a mean-field model. The answer would be known and it would serve as a benchmark (i.e. can your QC find it?). Such a simple problem is not meant to show any 'advantage' which remains arguable for the 2019 supremacy paper too. Just don't ask what 'advantage' means.
I would return your 1000 qubits and tell you that I would be happier if you gave me 100 qubits with 1000 low-error-depth. On such a device I would run VQE with a highly customised (variational Hamiltonian) ansatz to find the ground state energy of the Hubbard model at any filling.