I would think it would turn over if it was a kill switch issue.A low battery would at least get a click out of starter solenoid.Iwould check for power at the starter.I’m guessing it’s bad starter or the shift linkage is out of adjustment.
The kill switch will keep it from turning over. I’ve had that happen out on the lake. Six miles out and ready to head in at the end of the day and nothing. Motor won’t turn over. After a bit I figured out that somehow the kill switch had been triggered. Flipped the switch and away we went.
Felt pretty silly about that for a while. I still hear grief about it. “Is the kill switch on?”
OP, you have already heard a lot of good advice.
I have a 2014 Merc 150 4S. These motors drain batteries when they sit idle. Opening weekend my alternator died. We fished as long as we could without turning off the motor. Back at the dock, with a battery below 11.5 it would not turn over but fired it back up using a jump pack.
My take, if it wouldn’t turn over with a jump then something else is probably wrong. I’d start with resetting the computer by disconnecting the negative post for a few seconds. Also start with a fully charged battery reading 12.5v – 12.6v at the posts.
If that doesn’t do it, I’d skip all the other stuff and take it back to the dealer. Might be a starter solenoid, might be a wiring issue, might be a bad switch somewhere, who knows. I’d let them figure it out.
Also, your description of the starting procedure sounds correct. Turn to run and let the fuel pump pressurize, then turn to start and let go. From that point the computer will take over and run its own starting routine.