How then can coins come into the ecosystem?
You need to know what genesis block is all about. In 2009 the genesis block (block 0) was mined by Satoshi Nakamoto, at the time, there was no bitcoin trasaction time, transaction time was introduced in the block 1 which is 10 minutes on average. So, the only reward for the block 0 are the bitcoin that was mined.
In block 1 and the rest up till today and until all bitcoin are mined, miners receives two rewards, one from the fee paid for transactions and the other for solving the algorithm that create another block. That is just how the algorithm works.
After all bitcoin are mined, only transaction fee will be paid as miners reward because no bitcoin will be mined again. I hope this is explained in a way you will understand the differences between bitcoin mining and transaction.
Mining happens anyway, with or without txs. We need the inflation to stay approximately the way it has been structured to be through coding, the only variable being block time. If mining only happened once txs were initiated, that would mean there would be some blockchains with a temporarily fixed supply.. as in there'd be no more coins to be mined before someone initiates a transaction..
There will be transaction in every 10 minutes with the rate of bitcoin adoption normally. But you are right, transaction can not be a prerequisite for mining. But mining is a prerequisite for transactions until all bitcoin are mined.