Correct. You have 4 main control levers/valves. Boom, crowd, bucket, swing. 2 small levers, one on each side for the outriggers/stabilizers. Each valve will have 2 hoses to put pressure to one side or the other of the 2-way cylinders, depending on which way you move the control lever. The same hose will act as a return line when you move the lever the other direction, returning oil to the main oil reservoir internally though that valve body. If you have a wood splitter, or any other attachment with a 2-way cylinder, think of a hose at each end of the cylinder to make the cylinder ram move in the direction you want it to move by pulling the control valve lever. Same thing, only you have a bank of 6 controls & levers.
You have one hose from the pump feeding that valve body, and a hose going out the other side to return to the oil reservoir. Oil from the pump side is moving through that valve body under pressure circulating back to the return side constantly. When you pull a lever, you're diverting that pressurized flow of oil to one end of the cylinder of the function you are performing, like when pulling the crowd lever. When pulling the crowd lever to pressurize one side of the cylinder, oil on the other side of the piston inside the cylinder is pushed back through the block via the hose at the other end of the cylinder to the return line though ports in the control valve.
Below is an exploded view of a bank of 4 control valve similar to what you have. The main spool at the bottom of the pic with grooves is what diverts oil inside the block passing by those grooves, depending on what position you pull the control lever, so the grooves align with a port inside the valve body.
In your case, when you start the backhoe, and oil is coming out the hose, something inside that particular valve body is broken, or holding that spool valve open to divert oil to the hose that is leaking. I was thinking there may be O-rings on the main spool like some, but apparently, it's a machine fit with no O-rings. So, either a poppet, or a spring is broken, or a piece of debris like a piece of O-ring got lodged in a groove of the main spool is holding it open to allow oil to flow, pressurizing that particular hose. Merely speculating on what it could be.
All that to say just replacing that hose is not going to solve all of your problems. You have issues with that one particular control valve that is diverting oil to that cylinder. That will need to be addressed before making the backhoe operational again.
Just replacing the hose will make pressure deadhead against the pump, but "should" open the bypass valve to relieve pressure. I'd have to guess that happened anyway with pulling on the stump and when crowding, and it just stopped, it was bypassing oil. Old hoses will only take that so many times when coming up against maximum pressure until they burst.
Hope you can make heads or tails out of this.
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