In performance, the primary role of the conductor is indeed to provide synchronization and coordination, especially when there are pauses or tempo changes. The conductor often also helps players with entries which are preceded by long periods of rests, either as an alternative to them having to count hundreds of rests or as additional security, although in most cases in the classical repertoire the player should be sufficiently familiar with the work to know where to enter.
A good conductor also provides continuous reminders of the shape of the music, in terms of the required style, phrasing and balance, but these aspects, along with many other details, are usually primarily determined during previous rehearsals, and it is there that the conductor really makes his or her mark.
For small chamber music ensembles the synchronization is usually handled among the players, and in some cases this is successfully extended to small orchestral works, such as Mozart piano concertos which may be directed by the piano soloist. When the players are very familiar with a work they can often play it successfully without a conductor (and even without the music), but the result is likely to be more reliable and accurate with a conductor.
Experienced conductors often recognize that they are unnecessary in places, for example where the rhythm remains predictable for a long time and the set of players involved remains fixed, so they cease beating time for a while, merely using occasional hand movements to indicate various nuances. Some even deliberately cease beating time in certain passages in order to create some level of uncertainty, leaving it up to the leader of a section to define the precise timing.
(I play the violin in several symphony orchestras and have also played solo piano, violin and viola with orchestra on past occasions. I have also occasionally conducted orchestras in rehearsal and concert. However, most of my conducting experience has been with amateur orchestras, where the conductor has a somewhat more active job in keeping things together and has somewhat less scope for spontaneous musical expression during performance.)