I have never heard of that recommendation.  But I think it comes down to what is your ultimate interest.  This sounds like a good fundamental way to develop overall music skills and understand keys and chord structures.  Sounds like an interesting way to start young musically oriented types.  But if what you really want to do is specifically learn to play sax, you can start right away on sax.  

The piano-flute-sax path sounds like someone who wants to make a career in music and saxophone.  Not that you are old (much younger than me) but if you are just now starting, I think you should just have fun learning the instrument of your choice.

With that said, I do recommend that you get a good teacher and learn the sax with proper instruction.  The good news is that if you find that music is your life, you can still take keyboard and flute lessons in parallel.

Just get a saophone fingering chart and start on saxophone.

Since you more interesting in learning saxophone, iwould recommend to go for directly into it rather than going thruogh all these procedures. The field of music continues attracting the attention of people of all ages, who often purchase a variety of musical instruments rent from music store for personal use. To get more information here, https://plus.google.com/102642981778688407461, regarding musical instruments rent or repair or buy, go through this Google page of webmusic. You can also go for attending music lesson if interested in.