'Would' as in 'he would eat too much ice cream' has a slightly different meaning. In the present tense, we use 'will' to mean that someone continues to something even though they know it has a negative outcome. E.g. "I have no sympathy that you are so tired because you *will* stay up all night playing computer games!" The example above is the same but it uses 'would' because it is the past tense of 'will'. So the meaning is that he continued to eat too much ice cream (even though he knew the result would be bad).

"Would eat" is correct because it's a repeated past action. "Used to eat" is also possible, but "would" emphasizes the habitual nature. "Used to eat" is correct because it shows a past state (not just a repeated action). "Would eat" wouldn’t fit well here.