Korean lilacs: take out dead branches. Prune up to 1/3 of remaining branches. This is best done right after bloom. Doing it now may mean that you will not have flowers Next year as the tree is setting buds now. Suckers on bush lilacs are new sprouts around the base. They steal nutrients from the bushes. You can fertilize now with an all purpose shrub fertilizer available at garden centers. Fertilize very 2-3 years. See the following website for additional information on growing lilacs. See: https://extension.umn.edu/trees-and-shrubs/lilacs

I am still not clear on what part of the lilac with continue to grow after cutting out the suckers. What produces the next viable, flower-producing parts of the plant if you clear out ⅓ each year for 3 years. What's left?

I have read through your questions and the responses and watch the video.  I can see why you are confused.  First of all you should have removed 1/3rd of the oldest, thickest stems right to the ground.  Not 1//3 of ALL the canes. Then there are younger, not as thick stems coming up.  They are the ones that will be blooming.  When they referred to suckers in the video, they were talking about the multitude of skinny, twiggy growth coming up.  You remove a good portion of them allowing some to become thicker and actual stems as they grow.  You don't want to remove all of the new growth but you also don't want a jungle.  It is easier to see than describe.

Thank you for the clarification. I did only remove ⅓ of the oldest stems, but these lilacs have not been pruned for 30 years and are overgrown and 10-12 feet tall. So, I took out ⅓ of old growth this year; according to the video, I do the same thing again next year (⅓ of old growth). It will look pretty sparse by the end of the next pruning season, but I should expect some of the new growth will take over?

Yes it will.  Big job but worth it.  

Last questions on this topic, I think...I read the MN extension info and note that there is a mention of cutting Korean lilacs down to the ground in late winter. Mine is so large it is blocking the sidewalk to the front door. Is this a step you would recommend? And what is "late winter"? 
You also mention a all purpose shrub fertilizer--should I be looking for any particular contents and % of nutrients?
Thank you

That is a prime example of wrong plant/wrong place.  Yes, you can cut it back and make sure to prune it every year after it blooms to keep it small.  Late winter is around March.