The simple answer to your question is that the history of public showing of movies actually pre-dates the history of audio amplification. When films first began showing to the public in the 1890's, the amplifier had yet to be discovered, much less commercially available. Guitars, pianos or the Mighty Wurlitzer were significantly louder than the passively amplified Victrola.

The first stand-alone theater was built in 1899 (in Philadelphia) and it would be eleven years more till Edison invented the Kinetophone, whose purpose addresses your question. The Kinetophone was designed to record and play sound synchronized with the Kinetoscope - however the problem of amplification for a theater still remained.

By 1900, Edison had completion with the Phonorama, Phono Cinema Theatre and the Chromophone. They were all sound on disk systems built to do exactly what you had suggested in your post... to synch a record with the movie. They did not succeed. First of all, the recording time on a disc was only about 4 to 5 minutes. With movies getting longer, the opportunity to loose sync drastically increased with the length of the film. Secondly, it only took one scratch or one jump of the needle to take the film and audio out of sync. And finally.. even with double or triple amplification horns, they were still not loud enough for a reasonably sized theater.

It wasn't until 1920 that audio was actually added to the film itself, thus solving the synchronization issues. And as luck would have it, there were also strides finally being made to solve the amplification issues.

In 1906, the Audio Tube was patented by Lee de Forest, a contemporary of Tesla and Marconi, and is known as "The Father of Radio". However, it wouldn't be till 1922 that his first efforts of tube amplification, along with others in the development of vacuum triodes, would become useable enough to create a theatrical film amplification system. Interestingly enough... even though several theaters in New York adopted the technology, Hollywood did not.

Moving separately, but along similar lines, progress was being made on amplification systems for venues other than theaters. The Automatic Electric Company developed the first public address loudspeaker in 1910. By 1913, Comiskey Park in Chicago was making announcements and playing music during the ballgames. Along with them, Magnavox and Marconi were both developing loudspeaker and microphone technology for large venue. Magnavox developed a 25 watt amplifier to make a complete sound reproduction system. Marconi took the path of future rock concerts and developed a 200 speaker, 20 ton system to be used at public events.

By the early 1920's we had the first portable sound systems that could be used in theaters for either live or recorded music. What was not in place was an affordable, accepted and licensable method of synchronization to film.... remember the film industry survived to date on license payments from theaters, not necessarily adopting available technology.

By 1925, sound on disc synchronization reared its head again by virtue of the Vitaphone Company. Vitaphone received the same cold shoulder in Hollywood that de Forest did a few years earlier, with the exception of one studio... Warner Brothers. It should be noted here that Warner originally had zero intention of adding talking to their films at this point. They simply wanted a better method of providing music to their films. The problem that Hollywood had after the premier of Don Juan, the first film that used this process, was that it was a gigantic hit. Talking on screen in even a small part of the presentation was considered the "eighth wonder of the world". The rest of the movie studios were now worried as everything was changing.

When "The Jazz Singer" premiered.. it was all over for the old silent studio systems, actors with bad voices and all the local musicians employed by the local theaters. By 1927, only the "talkie's" brought any crowds to the theaters and movie sound studios began being built out of necesity. By 1928, every studio was releasing talkies and silent film was quickly dying. All that was left was to decide on whose amplification system would be licensed to all the theaters in different parts of the world.... which is another story in itself.

One side note - These breakthroughs happened right before the Great Depression. Were it not for two bold smaller studios early adoption of the talkie technologies (Warner and Fox), I suspect we would still have silent movies well into the 1930's.