I am always pulling up seedlings of Prunus laurocerasus in our woodland from plants/hedges in the area that have been left to bloom and set seed. Please be conscientious and do not let yours set seed, or prune off the bloom or seeds. Here is a link that might be useful: weed of concern

Don't do that with Epsom salts. It won't kill the laurel unless you render the soil so toxic via literally salting it that nothing will grow. However, it's far easier to throw the magnesium balance of the soil off and render the soil tight and nasty (magnesium excess tends to dispel some calcium and throw off the calcium/magnesium balance, rendering soils tighter, easy to crack, harder for water to penetrate...and it's not easy to reverse). The laurel will probably manage, being mature, but anything new won't like it. Other nutrients will have some difficulty absorbing as they're now competing with excessive magnesium as well, particularly if you use enough to damage or kill the laurel. So there's a second "no" on blind Epsom salt additions. If you're going to kill it, spray it with Round Up or another herbicide designed to kill plants. Or dig it out, or cut it at the base and remove as much root mass as possible. Or leave the roots, kill sprouts as they arise, and wait for the roots to die and rot a bit. Or use a stump killer. There are a lot of possibilities.

Haven't dealt with cherry laurels, but I haven't had anything come back after digging out roots to one foot deep, and that includes a Portugal laurel. The larger main roots may have to be dug deeper if they would interfere with your future plantings and cultivation.

"Wet in winter" is somewhat meaningless :-) Pretty much any PNW soil is wet in winter due to significant rainfall amounts however that does not necessarily limit what can be planted here, provided there is still good drainage. How wet IS wet in winter? Does the soil drain well? How long has the hedge been planted? Laurels are particularly robust plants that generally thrive with minimum attention and prefer moist soils. They dislike drying out and the potential wide swing between moister winter soils and drier summer soils can cause stress. Regardless if too much or too little moisture, it can take time for effects to become highly visible and the plants start to fail.