This Script adds every song in the playlist you're currently viewing at https://play.spotify.com to "Your Music". Open your browser console and paste this script to run it.

spotify-save-all-in-playlist.js

/*
 * This Script adds every song in the playlist you're currently
 * viewing at https://play.spotify.com to "Your Music". 
 * 
 * Open your browser console and paste this script to run it.
 */
(function() {
 
  // Check this web browser has all the functionality we need to do this task.
  var hasQuerySelector = typeof document.querySelector === 'function';
  var hasQuerySelectorAll = typeof document.querySelectorAll === 'function';
  var hasContentWindow = document.createElement('iframe').contentDocument === null;
 
  if (!hasQuerySelector || !hasQuerySelectorAll || !hasContentWindow) {
    return console.error('Please try a newer Web Browser.');
  }
 
  // Look for the playlist we should be viewing.
  var playlistFrame = document.querySelector('iframe[id^=collection]');
 
  if (!playlistFrame) {
    return console.error('No playlist found on this page.');
  }
 
  // Get the ID of the playlist you're currently viewing at https://play.spotify.com.
  var playlistFrameId = playlistFrame.getAttribute('id');
  
  // Get the child window that playlist is being displayed in.
  var playlistWindow = window[playlistFrameId];
 
  if (!playlistWindow) {
    return console.error('Cannot access playlist.');
  }
 
  // Get all the + or √ buttons
  var saveButtons = playlistWindow.document.querySelectorAll('[data-ta-id="track-add-button"]');
  
  // filter out the √ buttons (delete the "!" to filter out the + buttons instead).
  [].slice.call(saveButtons).filter(function(button) {
    var tableRow = button.parentNode.parentNode;
    var isTickButton = tableRow.className.search(/\badded\b/) !== -1;
    return !isTickButton;
  })
  
  // Loop over the remaining buttons.
  .forEach(function(button) {
    
    // and click them.
    button.click();
 
  });
 
}());

Any errors, please comment here and mention what browser and version you're using.

Note: To make this work for artist pages etc, use Spotify's Add all to playlist feature and convert the playlist.