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Well, your models are wrong. In both examples, you assume exponential growth will continue forever. Resource limits are a thing in the real world, as evidenced by every population in history (humans or animals).
I assumed linear growth and I assumed a frankly absurdly low growth rate. Under this model earth is only sending out a colony ship once every 15,000 years. Does that seem likely? We made 6 trips to the moon and about 50 years later are now planning more. Is there a single thing you can mention that humanity would only bother doing once every 15k years?
What is far far more likely is waves of ships, pauses of a century or less, more waves of ships...
Additionally I didn't assume forever. I assumed a malthusian growth pattern where by aliens keep growing until nothing is left.
Yeah, you assumed no catastrophic failures. On long timelines there are going to be world or civilization ending events.
There are so many species that were wiped out through their actions or just naturally. That's the point of the Drake equation; the sky should be full of other civilizations, but it isn't.
The common answer is that there may be a "great filter", some event that all advanced species encounter. Maybe it's ahead of us, or maybe it's behind us. It could be something simple like "walking upright is rare" or it could be some powerful weapon everyone discovers.