A pull-down menu to select either age or year of birth is immensely user-unfriendly. A menu should never have 50+ values on it, and doubly so when the values are numeric. This selection requires that the user pay a lot of attention to the menu, and is error-prone. Worse, since the error is likely to be only wrong by a year or two, the survey respondent isn't likely to correct it because it's close enough. This reduces your choice to two: buckets, and manual entry.
One consideration in selecting how to answer this question is to consider whether your survey respondent will be primarily answering your survey via keyboard or mouse. Making the survey respondent switch back and forth between modes of entry is more likely to result in them quitting out of the survey altogether. If you are going to switch between modes of entry, do so thoughtfully, and consider ordering your questions so that you limit the number of switches between them.
It's worthwhile to note that survey respondents who are on a mobile device tend to have higher response rates if the amount of keyboard entry is limited. If your game dev is mobile, or if you anticipate that a large number of your respondents will use a mobile device to respond, I would choose buckets over manual entry.
Demographic questions are personal questions, and people can have unexpected reactions to being asked personal questions. Requiring that someone divulge their exact age increases the likelihood that your survey respondent will either lie (if they're invested in the survey) or simply quit the survey (if they don't). The age range buckets decrease your chance of the respondent quitting the survey because they don't want to divulge information that they feel is personal.