I did a similar upgrade since I only play esports games at 1080P low like Val, CS2, etc. I went from a 5600X -> 7800X3D, and I kept my RX 6600 gpu. My old setup was probably roughly equal to your 12400F/3060 setup, and I have a lot to say on this upgrade type. I'll try to organize my opinions below, since I yap a lot. Read whatever you find useful, but I have many thoughts / experiences on this.

Perf Upgrade (objective FPS numbers)

The maximum theoretical benefit, can be upward of 70 to 100% in pure CPU bound scenarios. Thing is though: your 3060 needs a lot of help reaching that, even in "CPU bound" games. You will need to optimize to death - I'm talking like basically lowest graphics all the time, and GPU OC if you are really trying to min-max.

For the games I play: My benchmarks showed roughly 700 AVG / 300 1% low in CS2, and I seem to get 600 to 700FPS AVG in Val after all in-game tweaks. My originally setup was maybe 70 to 80% slower, as I remember it was 350 AVG / 160 1% for CS2 on my 5600X. I will say, the GPU bottleneck is technically "visible" sometimes. In-effect however, it's the difference between 700FPS avg (on 3060) and 900FPS avg (strong card) while minimum or 1% FPS stays the same. So, basically, a non-perceptible difference.

Also, I am less familiar with Fortnite / FC 26, but the pattern should be similar. AFAIK, you will still see FPS uplift, but those games are generally more GPU-bound, so the "70 to 100%" number might look lower for those. Fortnite is similarly CPU-bound at lowest settings, but FC 26 seems it would be GPU-bound on a 3060 (I could be wrong though). Take of that what you will.

Subjective Experience (my opinion on the upgrade)

Honestly? Both noticeably better and underwhelming at the same time. At the same HZ, games do always feel consistent, but it's not life changing. In my opinion, I would say you NEED to upgrade your monitor to have your new higher FPS = HZ, or else it's mostly deadweight. I had kept my 165Hz monitor, and although my old setup did drop below 165fps sometimes, it wasn't as bad as I made it out to be, so the 7800X3D didn't do much. I have the old 5600X PC with a 1660, and it's not that much worse feeling when both setups run at 165Hz. If you FPS is greater than HZ like 90% of the time, it's good enough.

In the end, I lucked out buying a 360Hz monitor, and I am happy feeling the extra FPS now. The gameplay experience is definitely way nicer, though you may be less excited depending on how much you care about these "little things" while playing. It seems you have a 240Hz (correct me if I am wrong), so the 7800X3D might be overkill if you choose to keep it. You may be able to get away with a cheaper CPU upgrade; I'll suggest it in the next section.

Verdict / Measuring it's "Value" (advice on your purchase)

Overall? It's a specialized upgrade, you need to know that before going into it. The value proposition is really terrible by 'most standards', as the FPS/$ is really low. Someone who plays 1440P AAA games will have to spend WAY less trying to achieving a similar 70 to 100% FPS uplift, from say a 3060 to a 9060XT. Basically, you pay the premium for almost building a new PC, and for the fact CPUs don't scale their benefits for cheap.

However, if you live and breathe these high FPS Esports titles - it is objectively the best upgrade for the lowest cost. You do also have a "new pc" which can slot in a GPU upgrade later. So, if you plan on keeping the PC for a while, it is a 'more worth it' long-term investment for your entertainment. Maybe you could get a 1440P 1000Hz monitor with your 11800X3D and 6080 in 2030, or whatever it is by then, lol.

If this situation seems unlikely, consider switching to a 14600K/KF for ideally $300 to $350ish. It will "get you over the line" for reaching a stable 240FPS more often, unlike your current 12400F. You will be left with probably like $800 more to spend on whatever else you want. From what I've read in the comments, I can say as a fellow working teen, you may be happier buying other goods. Personally, I invested in my health through sleep gadgets, meal prep, and some supplements, after running maybe 1100 out of my original 1500 savings on my PC setup. I was going to get a 9060xt but I skipped it after realizing I was already more than happy with what I had. The left over 400 made me realize I enjoyed other things at the time, even if I was really focused on getting that good PC. Consider what you else you like, you're probably better off at least "getting a little bit of everything". It might not be a factor if you budgeted very intentionally, but if you are "budget-conscious" try your best to not overspend on a few extra fps. The <$400 14600k upgrade would be my first choice, but if you go 7800X3D you could sell the left over parts, and make spending $1200 easier to stomach, maybe like $600 net cost

Anyways, sorry for the long yap, but hope what I said helps :)