My take on long tones: As a trumpet player; As a physician that taught muscle physiology for 27 years at a medical school.
As a trumpet player. I studied for several years under Eugene Blee, the longest running Principle Trumpet player for the Cincinnati Symphony Orchestra. He was the individual that drilled into me the importance of warming up with long tones. He taught me this skill in the 1970's. I still use this today, almost 50 years later. I use his exact routine before I start ANY rehearsal or performance when time is under my control. Why? Because I have been very successful when playing for an audience for these 50 yeaars. The RARE times I have not been able to use his warm up because I have arrived late at a performance, I fatigue early into the performance. So for me, those long tones are essential to get warmed up so to make it through even the most demanding performance.
As a physician teaching muscle physiology. Long tones gets the muscle fibers to align to a performance strength demand while gently providing blood flow to the muscle fibers. That stretch coordination is vital to keep the motor units to function optimally as a group in series that will minimize fatigue. Blood flow is essential, not only to get oxygen into the muscles to allow as much aerobic combustion with optimal ATP production (the bottom line fuel source for muscle [ATP is NOT only the racer's edge... also the Trumpet player's edge]) . THEN there is the exhaust. Once this oxygen burns, phosphate bonds are transferred and consumed, carbon by-products begin to form. The most stress causing carbon by-product is lactic acid. Lactic acid is removed and eliminated by the liver, but you got to get it out of the muscle and delivered to the liver to prevent a back up in the muscle AND for the liver to convert that lactic acid back to glucose to recycle that fuel source BACK to the muscle (This is called the Cori Cycle). That is the second essential component of blood flow, it gets the lactic acid out.
So putting it all together: Long tones improve blood flow to muscle that is being recruited for performance. The blood flow gets oxygen in and toxic by-products as a consequence of this oxygen delivery out. So if we are using long tones EFFECTIVELY this process WILL optimize performance.