I just finished getting a Sears push with a Quantum B&S ready to go that was a gift. 6hp. Same problem as yours. Number 917.388380. Im not real familiar with B&S of this vintage but here is what I found.
You can run your 917 number at the sears website and find the parts schematic. Sears will ship the parts to your local store if you would like.
Primer system uses a gasket that goes between the air box and the face of the carb, part number 795629. This gasket must be good for the primer system to work. You could probably make one from gasket material if you had to. The air from the primer is forced through a duct which is cast into the plastic on the back of the air box. Pretty hokey really.
The air box is held on with three 5/16 or 8mm bolts. The carb is held to the intake pipe with two more bolts which I think were 7/16 or 10mm. Take a piece of paper and draw the linkage positions for the governor rod and spring. 5 bolts and its off.
After the bolts are out, pull the carb toward you and it pulls off the intake pipe. Reset the sealing O ring by pulling it toward you a half an inch once the carb comes off. If that O ring is cracked, you will need a new one. Pull the gas line. Next, the PCV hose pulls off the back of the air box. I found the gas line was rotted and replaced it. I used 1/4 inch fuel hose. There were no extra gas filters on this one that I was working on, just a strainer in the tank.
Once you have the carb off, loosen the brass jet on the bottom and remove the bowl. 1/2inch socket. Work over a basin incase anything falls. The float is plastic, and is held on with a pin. Pull the pin and pick the float straight up off the carb. Put the pin and the needle in the basin. Shake the float near your ear, if there is gas inside you need a new one. When you lift the float off, the needle comes with it out of its seat. The needle rides in a recess that is on the float. Notice the orientation for reassembly. There is another big O ring that goes around the outside edge of the float bowl. If its cracked your going to need a new one.
Use a tag wire or a bread bag wire with the outside covering removed, and run the wire in and out of all the holes. Mine had the pilot jet plugged on the face of the venturi, which is right next to the primer air inlet from the bulb. If yours is like mine, there is a small brass plug with a really tiny hole in it that feeds the pilot jet. Because of the EPA, there are no float bowl vents on these carbs any more, and they use these pilot jets to control fuel flow up to the venturi. Give each hole a good shot of carb cleaner, and the wire treatment. Reassemble in the reverse of how you took it apart.
I found four holes in the bottom brass main jet that holds the float bowl on, of which two were also plugged with organic matter. The holes are so small, again because of the EPA, that almost anything that gets into the fuel system will plug the jets. Wire and carb cleaner to clean them out. Allow everything to dry and reassemble.
While your waiting, dump the old gas and rinse out the tank with a little new gas. Allow the fuel to flow through the tank as it normally would and catch it in a cup. Look for particulate matter, and flush until its clear.
Dont try to run the gas from last season. Dump it into your truck, and go buy fresh. Because of moisture condensation, gas in the small cans doesnt seem to last too long anymore, especially if it sits in the sunlight.
Put the carb back on and use a drop of motor oil on the gaskets and O rings for a good seal. Should fire right up when your done.
Good luck and if you have questions, PM me. Should take you about a half an hour. You can get a new air filter at Sears or one of the box stores. Part number should be B&S 5043 or Sears 491588. Spark plug is J19LM.