I have used quite a few bug/issue tracking systems and also task management and project management systems so far for professional purposes. Personally I am big fan of Getting Things Done, time and task management. For Getting Things Done, I also had my time with Google Tasks and Remember The Milk. Although they simply do the work, they have several shortcomings, as you mentioned before.
I have tried several things and within the scope of this question I will try to breakdown my experience with this tools based on your requirement.
Bugzilla
Bugzilla is a mature product which has several features, it is a great bug tracking software. You can easily create a demo instance if you want to give it a try. But based on your requirements, I don't think it is the best fit.
Redmine
I have played with Redmine, it is OK as an issue tracking system but I was not very satisfied with its usage. You can have a look at the demo page.
MantisBT
I also happen to play with MantisBT and in my opinion MantisBT is a barely OK bug tracking system and I don't think it will be the best fit.
Team Foundation Server
I use TFS on a daily basis, but mostly for version control and it is many things besides version control. Work items can be used as bug tracking and it a great tool for team management but it is not lightweight and I also don't think it is the best fit.
Wunderlist
Wunderlist is a great tool for GTD. It is very simple and list based. You can create several lists, you can create reminders etc. For personal use, this is my favorite choice for GTD, up to now. Although pro version has a few more features, it won't solve your problems based on your requirements.
Basecamp
I have used Basecamp, mainly for project management, task and issue management. You can create projects, and create tasks for each project. It is very easy to use and if you want to collaborate it is very easy. But it has its shortcomings.
In my humble opinion conventional issue tracking systems are no good for Getting Things Done. From what I understand, you need a simple yet powerful project management tool. Which brings us to our winner.
Trello
Trello is a web-based project management application developed by Fog Creek Software
Some of the features are:
- You can create projects. Projects are shown as lists.
- Create tasks. Tasks are shown as cards. You can also assign tasks to
other individuals.
- You can move cards easily from one project to another, i.e. from
Doing to Done.
- You can use tags, attach files and create checklists and etc.
It is very well explained in this short video : http://www.youtube.com/watch?v=xWiunIolf4s
Also @JoelSpolsky explains it in detail here : http://www.youtube.com/watch?v=aaDf1RqeLfo
In my opinion Trello would cover exactly what you need, most probably even more.