. . . what would the symptoms be of tree branches touching phoneline?
Assuming the sheath of the cable is still intact then the answer is "nothing".
1 branch has caused the cable to sag under its weight.
When there is a branch causing the cable to deviate from its natural (hyperbolic) curve, strain loading will be occurring at the two end of the cable. In the worst case the cable would either snap or be pulled from its fittings.
I have checked and there is no audible noise on the quiet line test although this was with a dect handest.
That is a good and sufficient test.
Would line stats show anything?
Only if the insulation of the pair has been sufficiently degraded to let moisture in.