I had a bonefide night mare with the ultra light 243 Winchester Ruger Mk II. I prefer the standard length sporter barrel.
Keys to the Kingdom on Ruger Tang safety and Mark II accuracy are simple and straight forward.
The center screw should only be tightened no tighter than you can tighten with your finger nail, I can not stress this enough. You can completely remove the center screw as the purpose is to adjust the height of the front of the trigger guard. If you pull the center of the action down with the center screw, your accuracy will be helter skelter and difficult to impossible to tune.
Next, free float the magazine box between the floor plate and the action. Again, the mag box should never put stress on the action. Drop the floor plate, stick your finger up in the mag well and try and wiggle the mag box. The fix, use a file on the bottom of the mag box or a belt sander.
Bed the action and freefloat the barrel before you ever fire a shot.
On the old tang safety rifles, push a tight patch down the barrel, if you feel tight and loose spots, replace the barrel or get rid of the gun. Of course, you can spend a lot of time and money trying to get a poor barrel to shoot well. Bullets with more bearing surface will work the best, but it is a Wild Guess as to what will work.
Custom barrels on Ruger 77 Mk II shoot extremely well. I have had custom 308s with Hart barrels and a Ruger 77 Mark II with a 26" Brux #5 contour in 6.5x47 Lapua that shoots bug holes.
The young kids in our family get started with Thompson Contender Carbines in 30/30 and or 7/30 with reduced loads. These rifles are very light and easy for a youngster to swing, and every youngster has shot them well. Remington Reduced loads in 30/30 kill like a bolt of lightning.
The break open action is also easy for the youngster to manipulate, and the Triggers on the TC's are adjustable to very light. I have one young nephew. When he was 5 years old, he sat in my lap in the deer stand with his TC in 30/30. He shot 5 does his first year, plus a bob cat. He wanted to shoot every squirrel and Blue Jay that he saw, we got to teach him some rules on sportsmanship. He is in his late 20s now, still hunts with that TC carbine.
Youngsters have weak fingers, remember this on the trigger pull. They also need a very short length of pull, so have a sacrificial stock that you can cut for their length of pull. Eye relief is one heck of an issue for a youngster, set the rifle up for the youngster where they can throw the gun up to the shoulder where they have complete field of view without "hunting" for the cross hair or focus.
We tried the Rossi break opens, two out of 5 were acceptable in accuracy, triggers are a real can of worms, unlike the Thompson Contenders with easily adjustable triggers.
OP, 13-year-olds are not the same in stature and strength in their frames. A less-than-max load will allow them to master the rifle, provided you accommodate their length of pull. We used a load of 36g of IMR 4895 with a Sierra 85g BTHP for deer for this 14-year-old neice, which she shot very well. Her dad shoots right out of the back of his garage, so she shoots often, gaining experience and confidence. We did put a Gentry muzzle break on the rifle, which allowed her to shoot with incredible accuracy.
The first experience with a rifle for a youngster is something they never forget, so make it a good one regarding gun fit, trigger pull, easy target acquisition, and recoil. Two nephews got Model 7's with Gentry muzzle breaks on them in 243 as they got a couple of years older where they could handle the rifle easier, but we did cut the plastic stocks to their length of pull.
