I wouldn’t connect the p. control wire and power antenna wires together, they both do essentially the same thing, but the power antenna wire sends out 12v when the radio section of the head unit is active, this is not for turning on an amplifier and should not be connected to that, just bad practice to do that. All this does is feed remote turn on power to the antenna wire when not in radio mode, does nothing for functionality of anything and should not be done.
You originally seemed to think it did not have a factory amp. Now it seems confirmed that it does have one. This is where it’s a good idea to find wiring info for your amplified factory setup to understand what’s going on better. Connecting remote wire to antenna wire to cars blue wire feels like haphazardly guessing. Messing with 12v signals, such as the remote wire and power antenna wire can fry things if connected wrong. You may need to use the antenna power wire, but not just twisting it together with the remote wire and such, that's never going to make it work.
Looked like there were a couple or few stereo options for the 2003 4runner, so you really need to figure out which system you have and then source the wiring diagram / connector pinout to help remove guessing. You know you likely have a factory amp in there now, so try to research what it is and see if you can find wiring information. Maybe even find a 4Runner forum to sign up to and ask them how to figure out what OEM stereo system you have. Example: I got a free Bose OEM amplifier out of some Chevy, with a cut harness plugged into it, no idea what vehicle it came out of, I looked up the number sticker on it, found wiring info, and connected it to my power supply / speakers / line inputs to see if it worked, and bench tested it, works fine (was pondering using it for a portable Bluetooth speaker build, but found it is not powerful enough). Has a ton of wires dangling out of it so is super confusing if you don’t have the wiring info. My point is, find out which amp you have and or OEM system package, and try to find some real wiring info. The OEM amplifier makes things more difficult.
Lastly, look at how your Sjoybring stereo is wired, find it’s manual, look at the harness pinout and how it plugs into your OEM stereo harness wiring; where exactly the remote turn on wires were powering, confirm where it’s speaker outputs were going, power antenna if used... etc., its 100% functioning, so you can mimic its wiring to the Kenwood harness. If the Sjoybring is indeed working fine, something is wrong with the Kenwoods wiring still or it's been damaged from mis-wiring.
Hope you get it figured out at some point.