Going for the highest altitude possible is certainly no way to save fuel in every airplane I have flown (aside for maybe turbojet lears?) - in fact, its a downright horrible idea if you are trying to fly long range.
For the last few planes I flew, along with the current one, FL370 for a few hours followed by FL410 then later (maybe 6hrs later) FL450....etc..is the way to stretch legs.
Only dorks who like to take goofy FMS and Airshow pictures for the internet struggle to FL510 (hey wait that me!!)
Its the corporate version of the regional adventure called "lets 410 it dude"
If you are bucking a 100+ headwind and by climbing an extra 5 or 6,000 feet you can drop the headwind by a third or more, it more than makes up for the lose of speed. While your true airspeed and mach may be lower initially, your ground speed will be the same or higher with a lower fuel burn. Besides the speed will come back as you get lighter. So in the long run during that flight you come out ahead. And that is why sometimes it is faster not to fly a straight line. Work the winds.