Rope drag and anchor failure are potential issues of this approach. So what you are wondering about is whether it makes sense to save a couple of bucks attaching the carabiner directly to the anchor point. No it does not.
You seem to be just starting out. Whenever you are leading, you would do best to be with a more experienced partner who has the basic required equipment. For two reasons: you still need mentoring to lead safely, and you don't have enough gear. As you learn to lead safely, and gradually buy gear, these problems will be solved. It will take a few years.
BTW, of all my ice climbing gear, the runners and carabiners make up about 5% of the cost of gear. Ropes are expensive, boots are expensive, tools are expensive, ice screws are expensive- runners and carabiners are relatively cheap on that scale.