Well, as I see it, you have a few options.

1. Everybody in on ONE. Yes, good musicians can and do start at the same time on the 1. That's kind of the default way to do it. Absolutely nothing wrong with that kind of start. There might be a couple of milliseconds difference in the actual hit on one, but not enough for anyone to notice.

2. Pick up notes. You hear this all the time too. In country..."and a 4 and" is an accepted way to kick off a song. The pick up can be the fiddles, steel, guitar, drums, and quite often, all of them combined.

3. Partial start.... this can be on the one or a pickup start but it generally only includes the drums, or the guitars..... again this instrumentation lineup varies from song to song.... the rest of the band waits until the end of the verse, or some other strategically located musical mile marker to enter.

4. Vocal start.... with the instruments coming in at some point AFTER the vocal sings a line. (See Alan Jackson Pop A Top*) *actually used a pickup followed by the vox... then the band.

You can find my music at:
www.herbhartley.com
Add nothing that adds nothing to the music.
You can make excuses or you can make progress but not both.

The magic you are looking for is in the work you are avoiding.