Analyze the following conceptual sorting algorithm: It iterates through a list and for every element, it scans the entire list to see if there are any smaller elements, then swaps if found. This process repeats until the list is sorted. Identify the specific Big O notation for the worst-case time complexity of this approach. Then, propose a more efficient algorithm (such as Quicksort or Merge Sort) to handle a dataset of 10 million integers. Explain the trade-offs between memory usage and processing speed for your proposed solution in a text-based format without using code syntax.