you might be best served generating a random number within the ID range of the games resource and requesting by ID, if that turns out to be too much trouble you could create a local copy of just the guids which should be much more doable, and then pick a random one and do a single request.
edit: seems like you can search by date added, could do something like getting the latest added games id, generate random between 1 and that number and request by that id, and do a rate limited retry in case of id not found. Unless the db ids are very sparse should average at 2 calls per random game, and less if you're willing to cache the latest id for a period of time
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