Even if you use API for requesting the stream, when google detects an abnormal amount of request from it your account would still be detected as abusing the service. I'd still advise on investing for a streaming server instead, so you can have full control of it as well.
Or better, just use google drive as a backup instead, and upload the mirrors to other hosting services. At least this way, whenever you get your content deleted from one of the mirrors, you can reupload it from your google drive backup. This way, you won't be abusing your google drive account, and you have a backup server that you don't pay for as much as compared from getting your own unmetered dedicated hosting.