I've been hunting since the early 1970's. I've ran across exactly 1 hunter in all of those years actually hunting with a 30-30. Now everyone I know owns at least one, but no one actually hunts with them around here.
Yes, they kill stuff just as dead as they did in 1895, but so will a spear. Both the 6.5X55 and 7X57 predate the 30-30 by several years; the 30-30 was a huge step backwards in cartridge development. The 6.5X55 and 7X57 are ballistic twins to the, 260, 6.5 Creedmoor and 7-08 so it isn't just new whiz bang cartridges that are better than the 30-30. Virtually every cartridge developed since smokeless powder is better.
While the 6.5X55 and 7X57 went on to be used all over the planet, used by several armies, used for long range target shooting, and take every animal on earth including elephant, as well as being the foundation for a significant number of newer cartridges, the 30-30 was plodding along with mediocre accuracy as a 100-150 yard deer cartridge.
And FWIW, I own 6 different 30-30 lever rifles made by Marlin and Winchester. I like them well enough, but I don't confuse nostalgia with performance.
And yes, everyone should have a 30-30 and the youth rifle should be a good choice. I really like my 16" Trapper in 30-30. 2nd from the right. Very similar to yours.