It sounds like your trailer is operating exactly as it was designed. At that time grey water was allowed in most places to drain out on the ground or - you put a cap on the outside drain that has a hose fitting and ran that hose off the site or into a grey water drain line that some campgrounds had.
Your grey water is not intended to be allowed to enter the black tank and should not be unless your somewhere in a pinch and have to do so. Think about how you would then have to empty the system - you would have to close the black valve then pull the outside cap. When you did a mix of black/grey water will be in the lower line and pour out. The original manual shows collecting that in a bucket - then hooking up your sewer drain line and pulling the black valve to fully drain the system. Messy - messy.
So what is happening to you is the black valve is closed (as it should be) and you have the main cap in place. The only place for grey water to go is to build up in the drain line - after a few gallons it backs up into the lowest spot - the tub. This is the same set up we have.
I'm in a campground that has only grey drains (rare these days) so my black tank valve stays closed and grey drains from a hose adaptor into a grey water pit pipe. The campground has what they call the "honey pot" and will come around and drain the black tank for you - or you can hook up and pull the trailer to their black drain and empty it yourself. We only use the black tank for liquids so we generally need to empty the black tank 3 times in a season. In most state campgrounds in Ohio for example - you will have no drain lines and will not be allowed to let the grey water run out on the ground.
Unless you want to rip the belly pan off and cut the grey lines and plumb in a grey tank (I'll pass) - get a hose adaptor and portable grey water tank with a short section of hose. We have a 30 gallon with a tow hook so you can pull it with the tow vehicle to the campgrounds grey water drain and empty it.
According to the 71 owners manual you should have a 14 gallon black tank if your trim level is Land Yacht and 21 gallons if the trim level is an International.
1 Garden Hose Adpater SKU5463 - $21.95 : ODMRV , Out-of-Doors Mart
Thetford SmartTote and SmartTote LX Portable Waste Tanks - Product - Camping World