I'll address what you are doing.
- Let him fall by his own weight; we've sent incomplete deliverables because of him and lost potential customers. I have not stepped in to complete his tasks nor will I.
If you want to be a good employee, your job is to make the companies goals succeed. If someone isn't doing something and it is needed to deliver something and it is the best thing for the company, you do it.
This reads like resentment. Resentment is, to be frank, your problem. If you are letting customers be lost and bad quality or incomplete stuff to be shipped because you resent a coworker, the companies problems include you.
- I told him about these issues and he replied he was sorry and that he will try to use the internal tools, he also mentioned that he is available for a chat when needed, I told him it is not my job to chase him for answers as the norm.
If there is something you need to get done, and using a specific tool doesn't solve it, then you find another solution. Saying "it isn't my fault the tool doesn't work -- it is because someone else isn't using it right" is an excuse.
Yes, it can suck if someone isn't using the "right" workflow. But the goal of a company isn't people using specific internal tools, except insofar as that helps the companies other goals. The goal is getting stuff done.
Again, this looks like resentment. You resent having to bother him to get work done. Feel free to resent the person -- but that doesn't mean you get a by on getting work done.
- For the tasks that he was blocking me in, I started to just deliver them, he just blocked me from completing tasks without his approval. His approval can take from two weeks to more than a month usually. This is insanely slow for the industry I work in.
This is a reasonable step. The next thing is, if the task is urgent, is to work out how to get approval done faster. Or get a way to do it without approval if approval isn't adding value.
This might require spending effort talking with the person, which might not be something you want to do. But if the task is urgent, then that should be your priority. If it isn't, then it can wait.
You can document how hard it is to get approval and mention this to your manager, saying it is a roadblock on your productivity, or how long it takes to get approval being a problem. You should first bring this up with the person witholding approval to see if there is a way to do it faster.
- I told the guy responsible for planning tasks of the low throughput that I have, and he told me he would speak with my coworker. My coworker then came to me infuriated saying that he is available for a chat if there is something I need to tell him and that there is no need to involve others. This confirmed to me he wants to control the communication flow.
Yes, you should first talk to him about it. Find ways to make it faster, and only if those ways to make it faster are failing raise it to the next level. And, if it is taking a lot of effort to make it faster, you should mention that while working on making it faster, before raising the issue to the next level.
The goal here is to succeed first, not fail in a way that isn't your fault.
If it proves difficult to succeed - either too difficult in that despite efforts you are failing, or just a real pain to make it good enough - raise those issues with your manager. But first, raise them with your peer in a non-confrontational way, possibly repeatedly.
- Communicate with your peer outside "standard" channels. Possibly including pinging that the "standard" channel requires attention. Do this when it is urgent, and when it isn't urgent and no action has been taken after a slight delay.
Why? Because your goal is success, not failure-with-excuse. One of your resources doesn't work with the standard tools? Work around it, don't give up.
- If someone doesn't deliver a piece of a product needed, and that product has value, ask your manager for permission to finish it. If you have latitude to pick your own tasks, just finish it yourself without asking permission.
During planning, be realistic about actual workload your resources (including this colleague) can pull off.
Why? Because your goal is success, not failure-with-excuse. One of your resources can't produce fast enough? Bypass it.
- If someone is blocking you from completing tasks without their approval, get their approval. Determine what is needed for that approval. If what is needed isn't reasonable, say it isn't reasonable, and explain why it is urgent. If the problem persists, talk to your manager about how the approval process results in a bottleneck, and see if there is an alternative. This can include what value, if any, the approval gave.
Why? Because your goal is success, not failure-with-excuse. If a step is needlessly difficult, you don't just give up. You put in the work required to get through the step. And then you look for process optimization. But first you succeed.
- Talk with the coworker first about issues. If they aren't solved, explain they aren't solved to the coworker. Try their solutions, and explain the problem with them if they don't work. Any issues you raise to management shouldn't be a surprise to the coworker.
Why? Because the goal is success, not failure-with-excuses. The coworker is a resource and a problem. You fix the problem and improve the resource. The manager can only give that coworker instructions (which you are capable of doing) and discipline them (which you cannot). Discipline is not the desired goal for the company -- they want their resources to be useful. So attempt to solve the problem and escalate on the specific problem that the attempt failed at.
Then if they come back, betrayed, you can honestly say "we tried to solve it, it didn't work, so I mentioned it to my manager as an unsolved productivity problem. We should keep at solving it. How about Y?"
It isn't about getting the coworker in trouble, it is about solving that productivity problem.
Remember: it may be the case that the delays you are seeing exist for a reason you are unaware of. Focus on the productivity problem, not the person causing it.
When you are managing someone, then focus on the person. When someone asks for peer reviews, you can list the issues you have.