Google's own advice is that the disavow tool is meant to be used as a last ditch effort. What webmasters are expected to do first is to make contact with the backlinking site and request that the offending backlinks are removed from their site and only if the webmaster does not do so should you use the disavow tool.
As for what the result of disavowing links is you need to be careful as the disavow tool can negatively affect your ranking if used incorrectly. When disavowing links make sure that the offending links are links that you don't want coming back to your site for ranking purposes and that they violate Google's terms. Whether it will make Google happy or not really depends on how the backlinking site is affecting your ranking which is why this can be a very difficult thing to do and why you need to be exceedingly careful. If your ranking has been negatively impacted by the backlink you may see your ranking go up a bit but if it hasn't then you may actually see your ranking go down. Based on your question with your ranking going backwards algorithmically make sure that it was due to bad backlinks, then try to get those backlinks removed, and if all else fails then disavow the links, but if the backlinks where not the issue check through webmaster tools if there is some other issue that Google has identified with your site that you can fix.