Rejecting Postdoc Extension for Other Grant Management Opportunities

Or am I obliged to extend the contract so that the project does not get stuck in the middle?

Let's start with the easy question -- you are not obliged to extend the contract. A post-doc is inherently a time-limited position; both your funding agency and your university are (or should be) well aware that you could leave at any time. Your case is even more clear cut, in that the university deliberately structured your contract in such a way that they could cut you loose at any time. This flexibility cuts both ways.

I feel the guilt because I had the experience of having to take over other postgraduate student's work because they left.

This shouldn't induce guilt; rather, you have created a new project that can be used to give a new opportunity to someone else. Sure, this might not happen and the work might be instead forced onto an already-overworked student, but that is not your concern.

Should I let my supervisor know that I would like to move on to a different career path

Well, there are competing norms here. One (mostly in business) is that you should never express any thoughts about leaving until you have accepted another job. Another (mostly in universities) is that you should give as much notice as possible so that your advisor can make plans. Which of these you opt for will depend on your supervisor's personality. You should also check your contract to see what the provisions for leaving are -- renewing your contract for six months might not mean that you have to stay for the full six months.

I also stumbled on a few job openings for research grant management elsewhere, which I am interested in applying.

The way you have written this makes it sound like you have only the vaguest idea about what this career path entails. If this is the case, I would caution you against becoming unemployed too soon; your job search may take some time. I'm not sure which position you are referring to, but "contract administrator" positions are usually bookkeeping positions that are not a natural fit for an academic, while "program manager" positions are generally rather competitive and might not be something you would be certain to easily get hired for directly after a post-doc.

First, to be honest, I've never been in your precise situation. But I do have a 40 year career in academia behind me. I've been in good situations and bad situations. I stayed in a bad situation for too long once and it was a mistake. I had options that I didn't recognize at the time, to my dismay.

I you have options that seem better to you than what you have now, then take advantage of it. Do not pass GO, do not collect $200. Go directly to your best option.

There is little worse in your single life than wasting it doing something you'd rather not be doing unless there are no other options.

You are at a natural point to change your direction. Just tell your PI that you now need to move on. If they can't understand the value of it then it is their problem.

You only have one life to live. Live it fully. Live it happily.