He was fairly young and coming into a new league and culture when Ferguson signed him. If i remember correctly, the main issue, at first, was his concentration. He would produce fantastic saves followed by howlers. Ferguson, being the great man-manager he was, always protected him from the criticism coming from people on the outside looking in (press, pundits, ex-players etc.), but, when he felt De Gea hadn't made the desired progress, he would give Lindegaard the occasional start, just to keep David on his toes. There was also talk about him lacking the skill to command his area and, purportedly, his coaches (Chris Woods and Eric Steele, i think) were working on that. But, back then, De Gea was one of the players around whom the new side would be built. Ferguson was always very protective of the players he believed would help the club move forward. How De Gea himself interpreted SAF's arm around his shoulder, we don't know. His concentration got better with time, but the rest of the weaknesses pertained. I guess it was OK back then, since an ageing defensive line and midfield saw us defending very deep and very narrow during our last successful campaign in 2012/13. We still conceded numerous goals from crosses and set-pieces, but Evra and Rafael's inability to defend high balls and our undercooked zonal-marking were being blamed more than De Gea. Plus, we were still winning stuff back then.
I think the next big chapter was LvG who brought Frans Hoek as his goalkeeper coach. The latter, having already spent a dozen years at Ajax and having trained Valdés and Reina at a young age, carried his own ideas along at Carrington and he enjoyed LvG's full support. And after he left United, he, more or less, admitted that he wasn't exactly loved by De Gea. One of the reasons was that he preferred his goalkeepers to train in isolation. Previously, the GKs at United were doing 80/90% of their training with the rest of the players. The logic behind it was to train the defence to act as a unit. Hoek believed that individual training should make up the bulk of the training sessions because that's the only way a goalkeeper can weed out his weaknesses. By repetition and by not relying on his centre-halves to do his job for him. From what i know, this is the way most goalkeepers train nowadays. I also believe that LvG's tenure was the time when De Gea showed actual progress in his game. He was a better sweeper-keeper, he was more bold in coming off his line and he had improved massively in tête-à-tête situations with an almost Neuer-like ability to hold his nerve and not going down until the very last moment. After one season, we had the incident with the broken fax-machine and the rumours that he wanted to go to Madrid.
Next stop, Mourinho. Jose decided to bring Emilio Álvarez as his goalkeeping coach. He was De Gea's mentor at Atlético. You can make of this what you will. Álvarez claimed that De Gea wanted to leave United and Mourinho wished to appease him and make him reconsider. All the progress under LvG evaporated, but he still gave us two good seasons. He was one of the highlights of our infamous second-place (81 points) season under Maureen, which remains our best tally in the post-Ferguson era. When Solskjaer was appointed, he brought in Richard Hartis (another United man, who had worked at the club under SAF and had also followed Solskajer to Molde) as the senior goalkeeping coach. Nevertheless, Emilio Álvarez initially retained his position among the staff. He left when De Gea decided to cut ties with Mendes.