Long story short, they did just that.

In late 1914/1915, there were two principle obstacles to employing fixed forward-firing weapons on aircraft. The first, obviously, was the lack of reliable interrupter gear. Even after the Fokker Eindecker was introduced, the technology was prone to jamming/misfiring, and often had a severely reduced rate of fire because the mechanism was tied to the speed of the engine. Several fighters came out without interupter gear using some work-arounds because of its drawbacks. The second big obstacle was weight. Aircraft engines of the period were extremely underpowered, and the additional weight of a weapon and its ammunition would often push the limit of what the aircraft was capable of. Furthermore, the big/heavy machine gun would have to be placed somewhere on the aircraft outside of the propeller arc that could take the load of the weapon. These two obstacles were what stopped doing what OP suggested from the outset.

Now, for the workarounds, which came in two main flavors. The first was to put a machine-gun, usually drum-fired, on the middle of the top wing of a biplane configuration, firing above the propeller arc. This was by far the most successful, and several aircraft with wings capable of taking the load used this during the Fokker Scourge and beyond, such as the Nieuport 10/11. The main drawback of this approach was the limited ammunition capacity and difficulty/impossibility of reloading, depending on the design. Another way to get around the lack of interrupter gear was by using a pusher propeller, instead of a tractor. The Airco DH.2 was the best example of this approach, which had a further benefit of using the engine as supplemental armor/protection for the pilot. Finally, some aircraft had machine guns strapped onto the fuselage sides at an angle, firing to the side of the arc, such as the Bristol Scout.

As /u/MeneMeneTekelUpharsi says, several methods of armament were tried before the single or twin synchronised machine gun became the standard fighter armament; another alternative, later in the war, was the moteur canon, an engine mounted cannon firing through the propeller hub. The most notable installation was the Spad XII with a manually loaded 37mm cannon flown by French aces René Fonck and Georges Guynemer, though it proved cumbersome and challenging to fly, load and aim the gun so was only used in small numbers.

Synchronised guns might seem like a more complex solution than just mounting guns outside the propeller arc, but incorporating heavy guns (and their associated recoil) into wings was quite a challenge, adversely affecting handling. Twin synchronised machine guns remained standard fighter armament into the 1930s (on e.g. the Hawker Fury and P-26), and though the most famous British (Spitfire, Hurricane) and American (P-51, P-47) fighters of World War II had wing mounted guns, synchronised guns were still widely used. Early versions of the P-40 and P-51 carried fuselage mounted synchronised machine guns before standardising on wing mounted .50s. The Bf 109 and Fw 190 carried a pair of machine guns in the fuselage firing through the propeller throughout the war. The Bf 109 was not intended to carry wing guns at all, but difficulties with an engine mounted motor cannon meant the Bf 109E-3 fitted a 20mm cannon in each wing; once the problems were resolved the wing guns were deleted. The Fw 190, unable to fit a cannon inside the radial engine, had 20mm cannon in the wing roots inside the propeller arc; synchronisation was assisted by electrically primed ammunition. The Soviet Air Force also favoured motor cannons in the Yak series, and synchronised cannon in the radial engined La-5/7. Synchronised guns lasted almost as long as the propeller driven fighter.

Four issues with wing mounted guns in early aircraft.

1. Early warplane wings (WW1 into the 1920s) were flimsy affairs made of wood and linen. They would need reinforcing to take the mass and recoil of guns which would make planes heavier, slower and less maneuverable. The forward fuselage near the engine was a much stronger location. Later warplanes (1930s onwards) often had stronger metal wings that could take the load.

2. Large masses a long way from the centre of mass will affect maneuverability. Think of an ice-skater spinning, when they stretch their arms out to the side their spin slows down, when they tuck their arms close in they spin faster. Like this. (Go and play on an office chair and test this out.) A fighter depends on a high roll rate to manoeuvre rapidly (rotating along its length horizontally not vertically, but the principle is the same) so putting heavy stuff like guns as close in as possible will help this. Later planes were heavier with metal wings, travelling faster, so the relative mass of the guns was not as significant and did not have as much effect on roll rate.

3. Early machine guns usually needed to be user accessible for cocking, re-loading, clearing jams, etc. Wing walking in a dogfight is a bad plan. Later guns had these problems largely designed out.

4. Guns mounted outboard need to be angled inwards slightly so that the bullets converge at a set distance ahead of the plane. This limits their effectiveness if your target is closer or further away than this distance so you need a very good reason to put guns in the wings. Early planes had too many negatives (1-3 above, plus other comments) to overcome this disadvantage and make it worthwhile.