Will a french drain help the runoff ditch in front of my house?
It sounds like the grading of your ditch and/or yard degraded (pun not intended) or was never perfectly graded.

even a a berm of a few inches is enough to trap water in a puddle and saturate the soil. Especially clay.

Some time with a shovel and a laser level to grade the yard and ditch better would be more effective.

If you lack a laser level or the eye to find the high spots, you can instead mark where the puddles end and dig there to lower the soil to provide a runoff path for the water. Repeat until you have no more standing puddles after rain. Put the excess soil into the puddle itself. (pull up the grass sods first as needed)