I have to "cycle" my throttles too. When starting up, I put it from off to idle, then back to off, then back to idle. No need to go to full throttle.
You can tick the "Sync cockpit controls to HOTAS
controls on mission start" option (Options, Misc tab), but I prefer this off. I keep my throttles on the OFF position when I start the game, but if you connect to a mission with the A10's engine started up, you are essentially doing an "auto shutdown" procedure
That is why I keep that option UNTICKED, but on the downside, I have to "cycle" all my Throttle controls and switches the first time I use it. As for the throttles and the startup sequence, I always visually confirm that my in-cockpit controls have moved to correspond with my actual HOTAS
Hope that helps, and welcome to SimHQ and DCS A10C!