Lori - 

Snow on the Mountain is a plant that is challenging to eradicate. It can be done, but will take persistence, time and energy.  The plant spreads by it's roots, which can go deep and long and even very small pieces can result in a plan growing back.  

As you have found, digging it out is a long process that requires constant monitoring and digging. Note that pulling out the plants will most likely leave some roots intact and result in hardier plants that come back.  If you choose the dig method, if possible for the area, after digging you could try placing a cover of black plastic, carpet or heavy landscape fabric over the area for a season is recommended.  The idea is to "starve" the plant of sunlight long enough that it cannot survive.  You will need to monitor the edges around the covering for shoots that sneak out .  

Unfortunately, the vinegar and salt method (with soap as a surfactant) will not be effective as a control for this plant.  The concentration of acetic acid in over the counter vinegars is too low (5%).  Most commercial pesticides containing vinegar contain 15-20% acetic acid (and an applicator license may be required because of the strong concentration and harm to skin or your eyes).  In addition, adding salt to the soil degrades the soil integrity and impedes water movement through the soil.   And, just in case you think of pouring boiling water on the plant, research has shown that you may only see a "top burn" of the plant.

This isn't probably what you wanted to hear as an answer, but a Glyphosate product will most likely be your best option.  Be aware to not pray when pollinators are present during the day. Multiple applications of the Glyphosate product, (possibly combined with digging) will likely be needed.  When spraying, make sure to protect surrounding plants with a non-porous barrier (plastic, heavy cardboard, rugs, etc.).

You will likely see reduction with the above methods but you will need to keep an eye out for newly emerging plants and dig out well.  No new planting should be done in the area until you no longer see signs of the plant.  This may take a growing season or two.  I have included a links to a few articles below which you may find interesting.  Good luck!

https://hyg.ipm.illinois.edu/article.php?id=899

https://www.maine.gov/dacf/mnap/features/invasive_plants/aegopodium.html