Yes DOGE would be much better as an independent agency created by Congress with all the controls that comes with it. Itβs inherently a bit toothless as an executive advisory council without further increasing executive power which is dangerous.
So far firings were mostly just from the subset of executive agencies they directly control, for ex USAID is legal because it wasn't create by legislative branch like the hundreds of other agencies. They can only fire probationary employees in those, with the exception of 'with cause' firing - a newer executive power as of 2020 supreme court ruling.
But I don't think employment is the primary waste issue. Federal employment has grown relatively slowly compared to local and state administration.
Which is why my point was about doing macro spending analysis. Most of the waste is in gov cost-plus contracts/procurement (feds outsource everything which is why employment hasn't grown much), how agencies operate (large duplication, old systems, etc), and more generally the unwieldy 1000 page congressional budgets no one reads.