I've made many OTA bookings before, and I don't recall a time where I was not given a direct airline PNR, which allowed me to do essentially everything I wanted to, similar to a direct booking. Anyone with back office knowledge who is able to shed some light on this?
It's not correct that you can't get the airline PNR; any reputable agent provides this with the e-Ticket. And it's not correct that you can't "do essentially everything" online with that PNR - only a handful of airlines (such as Aer Lingus) have limited functionality on their websites for third-party/agency bookings.
However, you cannot call up the airline to talk about/discuss/change the ticket. When you decided to use an agent, you went to a middleman, other than the airline. Therefore, you must continue to deal with that middleman.
The main difference is: you paid your money to the agent. The agent may or may not have yet paid the airline. They "own" the booking, so they are responsible for it, until it goes to airport control.
Because you chose to use an agent rather than the airline, the airline has no reason to deal with you. Airlines will typically only deal direct with you once the flight has opened for online check-in.
If you paid money to a third party, even if it is ultimately bound for that airline, you have to deal with that third party.
You should only consider buying from an agent if the ticket is not available otherwise (i.e. tour package discount, error fare etc) or if there is a significant discount available for doing it this way.