Without running a node, just via terminal, can you build and complete the broadcast of a transaction without using a third party PushTx API? How do I connect directly to the network? Is it possible? What do you need?
Yes, it is possible. You need to have a software that can connect to nodes on the network and speak the P2P protocol. That software can then broadcast the transaction by relaying it to multiple nodes who will relay it to their peers and so on and so forth.
One such software is the submittx script writen by Wladimir J. Van Der Laan.