Actually you’re right, I misread the problem at first. I thought that you had observed yourself not dying 1000 times (rather than observing “heads” 1000 times), in which case you should keep playing.
Applying my style of analyzing anthropic problems to this one: Suppose we have 1,000,000 * 2^1000 players. Half flip heads initially, half flip tails. About 1,000,000 will get heads 1,000 times. Of them, 500,000 will have flipped heads initially. So, your conclusion is correct.