There are several reasons why your proposal wouldn't be a satisfactory solution to the problem.

First, the Germans were short of skilled shipyard manpower because of the need for men in the Army and other industries. There simply weren't enough German men and women to rapidly expand shipyard labor ranks when so many other industries and the Armed Forces also needed labor. The Germans could have drafted French labor, but their experience in French aircraft factories under German occupation, proved that the French (probably deliberately) worked so slowly that their productive capacity was only 50% of German labor efficiency.

Second, U-boats required great amounts of steel and rubber, both of which were in very short supply in German-occupied Europe. The Germans had to keep switching the allocation priorities of steel and rubber (and other scarce commodities) around during the war as conditions changed and this proved a very inefficient way to produce any kind of weapons.

Third, the German economy was stretched to the breaking point as the war progressed. There weren't any spare funds for "research" until Hitler desperately decreed a priority for "wonder weapons" research in 1943. This, of course, was way too late to produce much in the way of advanced naval vessels which take years to develop. The Type XXI U-boat, for example, wasn't a radical new idea in submersibles; it was simply a normal design which emphasized very large banks of batteries to increase underwater speed. But because of the haste with which it was designed, it was poorly designed and built and hence didn't have much impact on the war.

Fourth, the Med wasn't a good place to operate subs, even though both Britain and Germany tried to do so. In any case, Donitz was reluctant to send subs to the Med because he didn't consider it a decisive theater for U-boat activity. The North Atlantic was the area in which the decisive U-boat/Convoy battles took place. The German's main problem in getting boats into the North Atlantic is that they had to transit the Bay of Biscay to get there, and the British put up an effective 24 hour air patrol over the Bay to interdict the U-boats.

Fifth, U-boats couldn't be built in the bomb-proof U-boat pens and the British, and later Americans, bombed the U-boat shipyards, and factories making U-boat equipment. This bombing would have been extended to any new shipyards wherever they were established.

Admiral Donitz sure thought so. The problem wasn't just a lack of training facilities. The biggest problem was manpower, again. There just weren't enough men in naval service to provide the crews by the time expanded U-boat production was achieved. By 1943, there were 23-year old ex-naval pilots skippering U-boats. Most of the crews were very inexperienced and green crews were extremely ineffective in the U-boat war. Also, at one point almost a third of all commissioned U-boats were in the Baltic in training roles. This meant that they couldn't be deployed to the North Atlantic where Donitz desperately needed them.

To gain a good understanding of the problems the Germans had in sustaining even a minimal U-boat effort, I recommend Clay Blair's "Hitler's U-boat War" (Volumes One and Two). Blair was a WW II submariner who has written extensively about submarine warfare in that conflict.