Thank you for the info, and my apologies for any delay in getting you an answer. There is no set response time, depending on the time of day and what experts online at a given moment.
What you are describing is not a matter of syncing or programming. The remote is programmed to operate your tv and your tv only, from the factory. That cannot be changed.
So if some buttons work but others don't, that points toward a defective remote, meaning the electronic membrane switches under the buttons are starting to fail, which is part of normal age related wear and tear.
Lets try a quick test of the factory remote. Take the tv remote into a dim/dark room, and look at the infrared emitter at the front edge of the remote, as viewed through the DISPLAY (not the viewfinder) of a digital camera. A cell phone cam, web cam, security camera, etc will also work. Each time you press a button on the remote, you should see the emitter at the front light up, flash, or glow slightly. If you do (for those specific buttons) then the remote is functional. If you do not, then the remote is bad and would need to be replaced. I can help you find a new remote at that point if needed.