I think that a bill should be sponsored by a Senator. The current system works fine as is.
Yet, it doesn't. The system being sought to change is not the Senate itself, but rather the chronic inactivity of Presidential Administrations when it comes to legislative affairs. We'd be better off having a Parliamentary system where we elect a leader amongst ourselves, for all the good having Presidents seems to do. This Administration seems to take it's time in
signing bills, let alone actively pursuing their own.
Even when Purple State and I proposed our own bills, from time to time our bills would be modified before they got to the floor on a Senator's whim, and if I recall correctly, in one or two cases, they were not introduced at all. Where is the harm, for all the potential
benefit, in letting the Administration have a direct line?