Considering that the life cycle of the viburnum leaf beetle (VLB) includes that the VLB larva hatch in mid to late spring and immediately start feeding on their host plants and then crawl down the trunk of the shrub to the ground to pupate in the soil in late June through early July, it would seem that the active eating of the viburnum is already happening as early as April; nevertheless, it may be logical to cut down the bushes now before the soil is infected and the live beetles emerge. In a very short amount of time adult beetles will emerge and begin feeding on viburnum foliage once again. In fact, it takes as little as 8-10 weeks to go from hatching to adult. When they aren’t damaging plants, VLB adults are mating and laying eggs on the new growth of viburnum shrubs.

Another approach, which –because it is already May – may not work for you right now, is to remove and destroy egg-infested twigs that the adults lay, prior to cutting down the bushes; this is easiest to locate once the plant has dropped its leaves between October and April. Waiting until cold weather may be your other option.

Since this beetle is actively eating in its various stages, there seems to be slim timing margin for getting rid of the bushes and also getting rid of the beetles. Unclear whether treating the VLB is advised or not.  I would be careful to not compost any portion of the Viburnum bushes. 

The following links may provide some help:
https://extension.unh.edu/blog/2019/06/what-eating-leaves-my-viburnum
https://www.canr.msu.edu/news/keep_your_eyes_open_for_the_viburnum_leaf_beetles
http://www.hort.cornell.edu/vlb/manfaq.html

Treatment may still be possible if there are larva crawling through the branches.

Regarding if the root needs to be cut out: possibly not, as I am not sure if the beetle or the larva will eat at those. However, there may be larva and/or beetles in the soil. Because of this, I am not sure that I would want to plant new bushes, especially viburnum, in the same spot.  At least not for a while.

I cannot recommend the best way to dispose of the infected branches.

You may wish to consult a Certified Arborist about many of these same questions as such a person would be able to come on site and assess the appearance of the bushes. We suggest Certified Arborists because they must pass a test and must acquire continuing education. You can locate such individuals at http://www.treesaregood.org/findanarborist/findanarborist.aspx