Is not proving I own the checking account enough with them when I first started up?
This seems to confuse a lot of people. When you first open an account with any kind of financial services provider, your risk level is assessed. The extent to which you're required to verify at the outset depends on that initial assessment. So, yeah - verifying that you control the account from which funds originate might be enough for you to successfully open an account with a payment processor (all that is meant to prove is that you're not funding your new account through unauthorised access to someone else's account) - although it's rarely sufficient for you to open an account with a bank or a credit union.
Your risk level isn't a static thing, though. It can change with how you use the account and Ongoing Customer Due Diligence and Enhanced Know Your Customer requirements mean that if the way you use the account indicates an increased fraud/AML/CFT risk a financial service provider is required to obtain more information about you, the source of your funds, and the destination of your funds. There are many types of transactions which increase the risk level of a given account and what makes them high risk might not be your personal actions but the nature of the businesses with which you're dealing, the geographic source or destination of funds, and a heap of other things which flag a transaction as "suspicious".
Some institutions require a higher level of verification from everyone when opening an account so that they're not constantly having to suspend accounts all the time, but even if you have fully verified at a high level you can be asked to do so again at any time.
Because Bitcoin exchanges and payment processors move large amounts of funds through the banking system and operate a service which is inherently high risk in terms of fraud, money laundering, and terrorism financing, financial institutions impose on them strict KYC requirements. While they could use financial institutions in countries which are less AML/CFT compliant, that carries its own risks and would also likely slow down the processing of payments - it would work only if those funds never need to be returned to the financial systems of more compliant countries.