What if the yellow cross fades away (as now) but the chart remains centred on the search location without reverting to tracking the aircraft until some other action is taken (e.g. tapping on the green arrow that points towards the current position). Would this cure your problem Timothy?

The (slight) argument against it being persistent is that it will actually slightly obscure the waypoint/entity you're looking for.

The yellow cross which shows the result of a search is not persistent enough.

Do others find it a bit frustrating that it disappears so quickly?

At the very least, could the persistence be increased to, say three or four minutes? But I would prefer to see it persistent until either it is tapped, or another search is started or a long timeout (say 20 mins) is reached.