e.g. when I looked at a dp solution for a medium leetcode problem, I couldn't generalize to an easy problem without looking at the hints.

Learning from your mistakes is only part of getting better. You won't be able to generalize the answer from one problem to another by only looking at your mistakes.

I'm new to this whole competitive coding thing, but it seem a lot like chess -- something I am much, much more experienced in.

Now in chess you won't get much better if you only look at the mistakes in your games -- although this is something that you need to do to improve. Looking at the mistakes helps identify the areas in which you are weak, from there you can identify what you need to improve on and then practice in that area.

Say for instance you lost a game because you missed a fork. Well, this suggests that you should brush up on this particular tactic. Now keep in mind, if you solve a chess tactic that is rated 1800 that uses a sacrifice and a fork that doesn't mean you are going to encounter that exact sacrifice and fork tactic in your future games (or ever again for that matter) nor are you necessarily rate at 1800 -- however, you are training your brain to pick up on this tactic subconsciously. You become aware of the idea that such a thing can be done. If you are new to chess it's one thing to learn what a fork is, it is then another thing to recognize it in a different situation.. how good you are is in part predicated on how well you can recognize these patterns and how fast you are at finding them.

Similarly in coding, in order to get better look at what your mistakes are. If it is DP then do problems that involve DP and then do random problems were you must realize that DP is the solution. If you are making a lot of small blunders -- silly mistakes -- then keep practicing.

Another key takeaway here when learning something new/ trying to improve is consistency -- you must consistently do problems or else you brain will forget and you lose all that practice you've done.

How much important is Fenwick Tree for Competitive Programming?

There is a union of humorists in the comments above

After getting accepted for a problem think about what cost you the most time on that problem? Is it thinking, implementation, debugging, or any other thing? Then think about how you can reduce the time. Then think about when you are thinking about the problem which path gives you no solution and how you can avoid that path. After understanding others' solutions (from the tutorial or somewhere else) think about how you can go that solution logically without guessing from reading the statement. Smooth your thought path for that problem. Note your mistakes and when you make a mistake (new or old) you can read through all of the mistakes you have done so far as punishment (don't mind, just fun).

Your problems might have two causes, not understanding DP or not having enough thinking ability.

If you find yourself good on problems without using DP, then i suggest you to learn more classical model of DP, and then try to solve more DP problems. Because generalizing a method is not easy before you fully understand it.

If you find yourself bad at learning in analogy, that is probably caused by insufficient thinking. Try best to make yourself totally understand the algorithm, you can write editorial after you solve a problem, or share solution with your friend. Teaching others is also a nice way to check if you have totally understand the problem or not. Meanwhile I prefer USACO/Codeforces rather than leetcode. Problems on Atcoder are also very nice but it's not good to learn a specific topic.