IMO, Germans quickly (by the time they found themselves on the receiving edge of the Soviet and British artillery, talk from late 1941 on) discovered that their artillery park is second best on the battlefield. They were not happy with 105mm, they were not happy with 15cm, and they were eager to both much improve on these calibers, as well as to introduce the 128mm gun.
The British were also 2nd tier with artillery. There are a number of elements that make up an Artillery force.
1. Good guns.
2. Good shells.
3. Good doctrine
4. Good communications
5. Good transport
there may be others. The British had real problems with no 2 and no 3 and no 4 varied a lot in time during WW II.
I'm not sure why the Germans were trying to reinvent the wheel with the much modified 150mm howitzer while having the Czech 15cm K4 in their possesion already by early 1939.
That is something of a puzzle but it may come down to being cheap and trying to use existing tooling or there may be a bit of NIH going on? or something else? The K4 was not designed for horse traction and perhaps that had something to with it?
My reasoning is still the same, as it was on the beginning of the thread: trade the shell weight for range, without over-doing either the shell weight to be too small, nor that expectations for the range is too great - all in order to keep the weapon's weight manageable. Perhaps 13+ km for the 88mm, 15+ km for the '5 in'?
Artillery Generals think that artillery is like playing golf. The Generals need a wide selection of weapons (guns, howitzer, mortars) to handle every job just like a golfer needs 9-14 clubs to handle any situation on the course
Trying to use just 4-5 clubs means a lot of compromise.
Thus 88 gun-how is the mainstay instead of the 105mm how, and so is the 127(8) instead of 150mm how.
We do run into the 'cube law' and in artillery circles that means that the shell weight is proportional to the cube of the diameter. So is the weight of the tube and the mount/carriage.
Not exact but close. Next limit is the weight of shell that a single man can stuff in the breech end of the gun. This is generally accepted to be about 100lbs or 45kg at least on steady ground or large ship. "lively" ships need a bit lighter shells. Larger guns without powered assist need two or more men to get the projectile into the barrel. This rather explains the popularity of the 15cm/6in guns and howitzers among the worlds armies.
However weight of shell also has advantages and disadvantages. The bigger/heavier shell with range further with the same initial velocity. It will also hold a larger quantity (percentage) of HE per pound/kg of shell weight. British really mucked this one up with their cheap steel.
However large shells also means a slower rate of fire. So the actual weight of shell delivered per minute can be rather variable and this sort of leads us into rates of fire and heat and barrel wear.
Getting back to the cube law we can work out a figure of merit for the different calibers of
3in=27................1
3.5=42.88..........1.59
4in=64................2.37
4.5=91.12..........3.37
5in=126.............4.66
5.5=166.............6.15
6in=216.............8
We can also assume that for similar guns (barrel lengths and muzzle velocities) the the gun tubes and the mounts (similar traverse and elevation) are going to show a similar progression. an 6in "gun" of roughly 2000fps velocity and having 45 degrees if elevation and 60 degrees of traverse is going to weigh about 8 times what an upgraded French 75mm weighed (3200llbs X 8 =25,600lbs/11,640kg) and in fact that comes out fairly close to the French/US 155mm GPF of 1917/1918 at 25,905lbs for 2410fps velocity but only 35 degrees elevation. The 155mm gets a lot more range, only part of which is from the higher velocity. There are some benefits of scale but perhaps not as much as we think?
The 8.8cm gun/howitzer may be a good idea, The British bungled it a bit so perhaps it didn't get a fair shake but if you try to turn it into anti-tank gun you lose some of the advantages. The 5in may be trying to fall into two camps if you try for one gun. Soviets stayed with both Howitzers and guns not only during the war but long after. New weapons and not legacy weapons.
Captured Soviet 122 mm weapons (both cannon and howitzer) can be out-bored to accept German shells once the captured Soviet shells are gone (same as with Soviet heavy AAA that was out-bored to fire German 88mm ammo).
A lot of this may depend on condition of the weapons and difficulty of manufacturing replacement barrel liners vs different size shells.
AA guns have a much shorter barrel life than field artillery. How much trouble it was to make new liners vs setting up a different shell production line? It may also have simplified the fire control by giving similar ballistics to the normal 88 guns (and fuse setting).
Czechs bored out (relined) German WW II 15ch howitzers after WW II to take Soviet 152mm howitzer shells. They also added a sizable muzzle brake.
This does get us back a bit to the heat problem. Howitzers use rather small charges most of the time. This causes less wear and tear on the tubes and carriages. Long ranges and rapid fire can cause a lot of wear. The US had figured for the invasion of NWE that they would get around 10,000 rounds per 105mm howitzer barrel. They had some experience in NA and Southern Italy. Perhaps not a lot, this was decided many months before the actual invasion. They found by winter 1944/45 they were only getting about 5,000 round per barrel (higher charges/higher rates of fire) and they were starting to run out of spare barrels.
Both the British and Germans (and French?) in WW I managed to blow up a lot of their own guns due to wear and poor quality control of the shells. This
may explain their conservatism is service weapons (but not experimental) leading up to WW II.
This is a much more complicated subject than it may appear, much like fighter design
Every gun (and it's ammo) was compromise between conflicting requirements.
Best gun in the world doesn't do much good it it is stuck in the mud 40 miles from where the action is.
The Paris gun was an amazing achievement. Wither it was cost effective is whole different story.