I understand that it's probably a pretty good instrument in comparison to other instruments. But from a history standpoint, why did it become as popular as it is today, and why will I never hear a lute, tuba, French horn, or violin solo in a heavy metal song?
This is a fascinating question, and leads to an even more fascinating one: why isn't the standard instrument of western pop music the banjo?
The modern pop group descended from two main predecessors - the Jazz combo, and the bluegrass/country-western band. Jazz was more influential in the roles of the instruments - that's where we get the idea of the rhythm section and the lead section, a distinction that is less important in bluegrass, and wholly absent in old-timey (or mountain) music. In the dixieland jazz era, the banjo was the main rhythm instrument that could provide a harmonic background, i.e., play chords, and the reason it was popular is because the high twangy sound could cut through the brass instruments and be heard. Instruments that were lower in register were competing with a lot of other instruments for sonic real estate, and so were less popular. In fact, jazz musicians who played guitar often played a 6-string banjo tuned like a guitar.
As the small Jazz Combo became more popular, there were fewer instruments fighting over that mid-range sonic real estate, and so instruments like the piano started to become the default for the rhythm section. The guitar was too quiet to be heard until the invention of the electric guitar, and Charlie Christian essentially invented its role as a lead instrument in Bennie Goodman's small combo.
Meanwhile, the guitar was only one instrument of many in the bluegrass and country string bands, which usually had a banjo, mandolin, string bass, fiddle, and guitar. Bluegrass music was an uptempo and countrified version of the older 'old-timey' or 'mountain' music, which didn't use a rhythm section at all. Instead, if a banjo and a fiddle were playing together, typically they'd both play the melody, with their own embellishments. In the country bands, the bass would typically keep the beat, and the other string instruments would cover the harmonic side when they weren't playing a solo.
So what happened? Why did the guitar take over from the banjo, and become the main instrument of rock and roll, when it was never more than a second tier instrument in country and jazz?
The answer is that the guitar was one of the first instruments to be electrified. A banjo's high timbre isn't well suited to electrification, but a guitar sitting in the middle of the range, is. Once the unique sound of the electric guitar became sought after, someone figured out that rather than amplifying an acoustic guitar, you could just put some strings across a plank of wood, add a few pickups, and have a perfectly good instrument. The solid body guitar was cheaper, and so was mass produced (and vice versa). Suddenly for young musicians wanting to pick up a cheap instrument, the electric guitar was the best choice. And once it became popular, there was nothing else to really push it out of its dominant position.
Well assuming you are referring only to Western Music, the 5 and 6 string guitar has a large pitch range, and a variety of playing techniques, producing different timbres. Making it very versatile in tackling many styles of music, esp modern. The french horn, flute, and violin that you mentioned have a smaller range of pitch and more limited in timbre. When you combine all these instruments in a symphony or orchestra they compliment each other, but the french horn has less of a range in timbre . But when it comes to solitary instruments the Guitar and Piano are dominant not because they magical appeared as the Guitar or the Piano, but through a slow evolution and development of guitar like, stringed instruments and piano like key instruments over hundreds of years. Im sure a musical historian can provide a much better answer with references.
There's a large number of heavy metal bands that use different instrumentation.
The Swiss band Eluveitie use violin, bagpipes, flutes among others, with occasional solos.
The Finnish band Korpiklaani use the above, and also accordions.
Folk Metal and Medieval Metal throw in lutes.
I confess I haven't heard any tuba solos, although a google search of 'Tuba Heavy Metal' provides a large list of artists doing covers using 'the original heavy metal instrument'