Hey! Thank you for your question.
You are right regarding the sessionId value. Its purpose it to enable contracts deployed on different chains to coordinate by fetching the right messages from the Mailbox.
Would it not be easier to add some other access management scheme, e.g. add an ExecutionAddress/EntryPoint that can trigger the interop txs? The only problem then would be paying for gas, but that is simpler and can also be solved
This standard aims at capturing communication protocols where application contracts proactively fetch information from the Mailbox. We call this communication model PULL (the application pulls from the Mailbox). What you are proposing makes all the sense, yet it corresponds to the PUSH model where the Mailbox calls a function on the destination chain contract. This communication model is the most common in practice and there are already a number of standards for it, such as ERC 7786.