Were these airmen only freed up as the Allies destroyed more Luftwaffe aircraft and facilities, thus destroying their original air duties? Or was the Wehrmacht just horrible at people management?

A bit of both. Many eager Nazi Party officials, including the commander of the Luftwaffe, Hermann Göring, maintained their own "little empires" and contributed to infighting within the German government and military. As Prussian Minister of the Interior, Göring had formed an elite police battalion in 1933, which eventually grew into the division-sized Fallschirm-Panzer Division 1. Hermann Göring. It fought in Italy and Poland, compiling a good combat record in comparison to Luftwaffe infantry formations, but was accused of a multitude of war crimes against Italian civilians. By 1941, the Luftwaffe had grown to a strength of 1,700,000 personnel, and had absorbed many duties not directly related to flying aircraft; anti-aircraft defense consumed 35 percent of this personnel, and signals consumed 18 percent, while only 588,000 men, or 36 percent, served in flying units. The Luftwaffe was 20 percent of the strength of the entire German military in late 1941.

Heavy losses on the Eastern Front in late 1941 and early 1942 sapped the strength of the units fighting there. General der Fallschirmtruppe Eugen Meindl, who would later gain fame as a commander of parachute units in Normandy in 1944, commanded an ad-hoc formation of Luftwaffe, Heer, and Waffen-SS troops known as Gruppe Meindl (Division Meindl) in January 1942. Seven Luftwaffe "field regiments" were formed as an emergency expedient in early 1942, but the men were poorly trained, having only a few weeks of introductory courses.

It was proposed to transfer 50,000 men from the Luftwaffe to the Heer for retraining as infantry in the middle of 1942, and Göring confronted Adolf Hitler about this loss of his well-trained and loyal manpower. As a consolation, Hitler allowed him to raise Luftwaffe-manned infantry divisions under the command of Luftwaffe officers. A call for volunteers went out on 17 September 1942. In their first months of combat in Russia, the field divisions suffered heavy losses, and German military leaders questioned their efficacy. Many of their commanders, as well as their men, had little or no practical infantry experience, morale was poor, and they were short of the allotted equipment.

The Army remained unconvinced that they served any good purpose at all, and considered these units a waste of personnel and equipment. At least two major Army figures, Generalfeldmarschall Manstein and General der Infanterie Walter Warlimont of Army headquarters, had opposed the formation of the divisions from the beginning, but were overruled by Hitler. The fact that the organization of the Luftwaffe units postponed the refitting of four or five panzer divisions with new vehicles also vexed Army leaders. Their frustration spread among the lower Army ranks, who labeled the air force units 'Luftwaffen-Feldkonstruktions-Divisionen'--'mistakenly-constructed air force divisions'.

The failure of the Luftwaffe field divisions was a topic of discussion through the summer of 1943; it was proposed to convert them to more elite types of divisions in light of the quality of their personnel, such as luftlande (air-transported) divisions or "sturm" divisions which would conduct surprise attacks against Soviet forces in concert with panzer divisions. Parachute units stationed on occupation duty in France were permitted to draw on the divisions for replacements. In September 1943, Hitler authorized the Heer to assume responsibility for the divisions, severing the Luftwaffe's direct control. Many Luftwaffe commanders were summarily replaced, the divisions were converted to use the organization of the 1944-type Heer infantry division, and two field divisions were eventually entirely amalgamated, even losing their designations as "Field Divisions (Luftwaffe)." The original plan was to form 22 Luftwaffe field divisions, but only 21 were fully constructed; one was a re-formed Gruppe Meindl. The balance fought on the Eastern Front, although four saw service in Normandy (all were destroyed in combat or disbanded), two in Italy (both were disbanded), and one on garrison duty in occupied Norway.

Division	Fate
1. Luftwaffe Felddivision	Destroyed in combat, January 1944
2. Luftwaffe Felddivision	Disbanded, January 1944
3. Luftwaffe Felddivision	Disbanded, January 1944
4. Luftwaffe Felddivision	Destroyed in combat, June 1944
5. Luftwaffe Felddivision	Destroyed in combat, January 1944
6. Luftwaffe Felddivision	Destroyed in combat, June 1944
7. Luftwaffe Felddivision	Disbanded, May 1943
8. Luftwaffe Felddivision	Disbanded, April 1943
9. Luftwaffe Felddivision	Destroyed in combat, January 1944
10. Luftwaffe Felddivision	Disbanded, February 1944
11. Luftwaffe Felddivision	Surrendered, May 1945
12. Luftwaffe Felddivision	Surrendered, May 1945
13. Luftwaffe Felddivision	Destroyed in combat, January 1944
14. Luftwaffe Felddivision	Surrendered, May 1945
15. Luftwaffe Felddivision	Disbanded, October 1943
16. Luftwaffe Felddivision	Destroyed in combat, July 1944
17. Luftwaffe Felddivision	Disbanded, August 1944
18. Luftwaffe Felddivision	Destroyed in combat, September 1944
19. Luftwaffe Felddivision	Disbanded, July 1944
20. Luftwaffe Felddivision	Disbanded, November 1944
21. Luftwaffe Felddivision	Surrendered, May 1945
22. Luftwaffe Felddivision	Formation never completed

Source

Ruffner, Kevin C. Luftwaffe Field Divisions 1941-1945. Oxford: Osprey Publishing, 1990.