Is there such a service as an unlimited Bitcoin wallet with an API?
While I'm not sure of the technical requirements or limitations you have around building you platform. It sounds to me like your best option is simply running your own Bitcoin node itself. There are lots of online examples that could help you get started. I found bitcoin.org's guides particularly useful: 


https://bitcoin.org/en/developer-examples
https://bitcoin.org/en/developer-reference


This will also give you benefit of actually holding the keys to the funds on your platform as well.

For monitor transaction sent to your node's wallet, I'd check out the ListTransactions RPC call, and for broadcasting transactions, you can use SendToAddress, or CreateRawTransaction, SignRawTransactionWithWallet and SendRawTransaction.

I should also add that running your node can be almost free (just need a computer running the server), and that you do not need to actually mine BTC on the node (which consumes a lot of energy &amp; resources). Your node will just be monitoring the chain and broadcasting your transactions.