Have done both routes and I would qualify the pay point.
Through an agency, paying more mostly does not reach the person, so it buys you less loyalty than the number suggests. Direct, it reaches them and works as you describe. That distinction is worth knowing before concluding that paying above rate does not help, it depends entirely on who is receiving it.
What the agency does buy is continuity when someone leaves, which is real if your process is written down and near worthless if it is not. Which loops back to the first point: with documentation the agency is genuinely useful, without it you are just re-teaching a stranger every few months and paying a margin for the privilege.
I went direct in the end. But I would not tell someone with no documentation to go direct, because they are one departure away from starting over.