Fixed terms are in the Constitution, so when the original law set the starts of the first presidential and Congressional terms, those dates were then immutable. The Senate Judiciary Committee explained it thus, in Senate Report 26, 72nd Congress, 1st session (PDF p. 33):

The question is sometimes asked, Why is an amendment to the Constitution necessary to bring about this desirable change? The Constitution [before this amendment] does not provide the date when the terms of Senators and Representatives shall begin. It does fix the term of Senators at 6 years and of Members of the House of Representatives at 2 years. The commencement of the terms of the first President and Vice President and of Senators and Representatives composing the First Congress was fixed by an act of [the Continental] Congress adopted September 13, 1788, and that act provided 'that the first Wednesday in March next to be the time for commencing proceedings under the Constitution.' It happened that the first Wednesday in March was the 4th day of March, and hence the terms of the President and Vice President and Members of Congress began on the 4th day of March. Since the Constitution provides that the term of Senators shall be 6 years and the term of Members of the House of Representatives 2 years, it follows that this change cannot be made without changing the terms of office of Senators and Representatives, which would in effect be a change of the Constitution. By another act (the act of March 1, 1792) Congress provided that the terms of President and Vice President should commence on the 4th day of March after their election. It seems clear, therefore, that an amendment to the Constitution is necessary to give relief from existing conditions.

Also, the 12th Amendment hard-coded a date for part of the presidential term: "And if the House of Representatives shall not choose a President whenever the right of choice shall devolve upon them, before the fourth day of March next following". So if some law could change the start of the term, this deadline would not match that changed law -- but this deadline could be changed only by another amendment.

Also, Sections 3 and 4 of the 20th Amendment had another provision that only a constitutional amendment could enact: it covered a corner case when there would be neither president nor vice president and no way to elect them (if the election of both offices were being decided by their respective houses, and neither house managed to elect a person before the start of the new terms), and for death of presidential and/or vice presidential candidates.