Old drills randomly starts slow and rattling (if at all)

My grandfather gave me two old drills, as he does not use them anymore. The drills are the AEG 800W Powermatic RL (SB2E-801RL) and the SKIL 1473H2. They suffer from the same problem, although the AEG slightly more than the other. When pulling the trigger, they sometimes give their full speed as supposed to, but other times they start slower, and/or they start rattling, if at all. Then, suddenly (while keeping the trigger pressed) they might step up to a higher speed. It then keeps the speed if it is the maximum speed. If not, it might receive another speed boost a short moment later, until it reaches the maximum speed, but it does not slow down unless releasing the trigger. The behaviour happens more frequently when holding the drill sideways. The frequency of the problem does not seem to be affected by choosing the rotational direction of the drill head, or the instructed velocity. Although, for low commanded velocities, it more frequently remains stuck.

Since they are completely different models while suffering from the same problem, it makes me think that this is a side affect of their old age, which as far as I know is the only property they have in common. But, what exactly causes it, and can it be fixed?

My first guess is that this is caused by old rusted and/or worn down brushes of the motor, but to my understanding this should simply lower the power and efficiency of the drills as a whole, and not induce strange step-up behaviour. So far, I haven’t been able to find someone who has experienced the same behaviour with one of their (old) drills.

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Update: the answer and comments convinced me to check out the brushes after all, starting with the AEG. The brushes were more easily accessible than I thought. Upon inspectation however I didn’t see anything wrong. They didn’t seem worn out given that they were spring loaded with at least 1cm more of travel, and their surfaces looked smooth and did not show rust. Nevertheless, I cleaned them from all the dust and carefully polished the tips,ever so lightly (perhaps not enough), with the hope that this would solve the problem. Unfortunately though, it didn’t. The performance did not seem to be improved in any way. Below attached some pictures.