This might be an occasion for glyphosate if all else has failed. It only kills things through contact with leaves, so if you keep it of the leaves of things you want there shouldn't be a problem.

Wear rubber gloves and dip into the glyphosate before running your hand over the plant leaves. Spraying the stuff is not a good idea unless you want to kill off everything in an area. As this has been going on for a number of years you may need to repeat treatment but it is preferable to throwing away perfectly healthy shrubs. Welcome to GOY. If you want to grow Hout again plant it in a pot buried in the garden then it is much easier to restrain its growth.

Glyphosate does not kill woody plants (provided you don't buy a solution that's strong enough to kill woody tree stumps, though it would be difficult to buy that particular one in error). This means you can spray the Houttuyana with impunity, you just have to be careful not to get it on leaves of plants you do want to keep.

Look at it another way - it is providing good coloured ground cover keeping down the weeds and it can easily be cut back. It's unlikely that it is causing harm to the shrubs and could actually be benefitting them by helping the soil retain moisture - conversely, if it's really running rampant then it could be that you have very wet, poor-draining soil which isn't good for the shrubs.