2007 FLHX harley stalls starter backfires out intake but runs strong once started.

Sounds like a worn shaft bushing allowing the armature to rub on the field windings of the starter.

Question about 2007 Harley Davidson FLHX Street Glide
Peter, There is a Harley Tech Tip out for just what you are experiencing TT410. Attached links has 3 pages you can click through or download pdf.
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File preview TT410 pdf

Does the bike run if you give it throttle past idle? The motor should accelerate and gain RPM's when doing so. If it does and gains RPM's but is sluggish in response, I would check to make sure it has good gas flow. Check flow off the petcock, if good, check gas filter on the motor side. If still not running correctly, I would check the air cleaner inside to make sure some critter didn't get in there and make a nest or something. Beyond those ideas, I would bring it to a profession shop or a friend in the know.

Find and pull the starter relay then try it again. If the starter still engages, it's something in your starter in the solenoid part. If this stops the problem, you've either got a bad starter relay or a wiring problem in the start switch. Have you made any changes in the wiring recently?

Good Luck
Steve

to me it sounds like it might need some spark plugs put in

my roadking just did the same thing, check the gasket between carb and intake