I thought using an engine flush in an old motor, that hasn’t seen engine flush previously, would all depends on the conditions of the engine internals.
If it’s full of a mass of caked on gunk, then it would probably be a bad idea as freeing up large chunks of sludge may block internal passages. If it’s reasonable with no mass of caked on gunk it shouldn’t be a big issue. I’d think one could always check under the rocker cover to get a better idea of the condition of internal sludging
Some also say engine flush will also cause gasket leaks in an old engine. I’d think such gasket leaks are inevitable even when using a modern oil in an old engine as the modern oils have quality detergents in them (far superior to that in older type oils). And possibly these modern oils are not designed to work well with old style cork gaskets (since modern engines tend to use silicone).
Whatever the case, old tired engines can’t magically be restored by any type of additive. That’s what engine rebuilds are for
As to a milkshake sludge, not sure what would break it down while not being damaging to gaskets. Supposedly removing gasses from an emulsion helps to break it down. Best to clean it up as much as possible by draining overnight. It may even be worth using a heater blowing on the engine block during the night which will heat up and dry any moisture. Obviously such drying out will work better if the engine is not sealed so the crank case can breath by taking rocker covers off. You can always test the effectiveness of heat in removing milkshake on your radiator cap by using your wife’s hair dryer (just don’t tell her what you’re using it for). Then fill up with new oil and new filter and do a couple of long drives with frequent stops to check for frothing of oil. With a couple of rapid oils/filter changes you should be good to go as already mentioned.