Thaler and Sunstein called this "libertarian paternalism." Libertarian because no one's options are removed. Paternalistic because the choice architecture is designed with an eye toward outcomes that make people's lives better. The employee who is auto-enrolled in the retirement plan can still opt out the next day. The organ donor can change their registration. The green energy customer can switch back. The nudge is a suggestion with no penalty for rejecting it.
This mattered philosophically because it resolved a long-standing tension between two intuitions: the liberal intuition that people should be free to make their own choices, and the paternalist intuition that some choices lead to outcomes people themselves would prefer to avoid. Nudge said you do not have to choose between those intuitions. The design of the choice environment is the lever.
The influence has been substantial. The UK Behavioural Insights Team, founded in 2010 partly on the basis of this work, has run over 750 randomized controlled trials in public policy. Similar units now operate in over 50 countries. Thaler received the Nobel Prize in Economics in 2017. The question has shifted from whether to apply these ideas to how to apply them well.