To answer the main questions posed:
Yes, there should be an abundance of G1 - 3 materials (I dare say even G4 tbh) at considerably more opportunities and make them easy to come by. They're supposed to be common, not rare. Blowing up a ship in a CZ should drop considerably more of the 'common' G1 - G4 manufactured mats than it currently does for example, as should dropping into an emissions signal. Whether that's done by dropping more 'items' or making each item worth more materials is probably neither here nor there, although sometimes collecting the materials as they float into space can be a challenge in itself, especially in a CZ or pirate threat signal where you're being shot at.
I feel increasing the availability of materials as mission rewards would go a long way to fixing this grind too, getting 3 MEF for completing a mission that took an hour, is like offering 1 'buy three, get one free' tickets to see the Ghostbusters reboot from a few years ago, at the cinema, because you sat through the Oceans 8 film... give 10 for completing a mission that took an hour, especially if it's a hard one! Maybe offer 3 if your reputation with the faction is poor and 10 - 15 if your reputation is allied, something like that.
Although manufactured mats are by far the most annoying to collect (Because they're so heavily used in engineering and so mind-numbingly repetitive to gather and drop in such small amounts), I would consider dishing out encoded mats more regularly too. I dare say, that scanning a ship (any ship) successfully should dish out encoded mats readily. The whole thing of going to Jamesons crash site and relogging is horrid, but it's less horrid than just sitting outside a station scanning ships repeatedly for a decade to upgrade one module...
As for the idea of engineers dishing out materials, I don't think this makes sense from a realism perspective. Mats are their price for performing the engineering upgrades, so perhaps a slight twist to this, which could be more realistic and more engaging, would be reputation with the engineer (Separate from the current reputation mechanic tied to engineering level), earning you a discount on materials. This reputation could drop over time as well, so although you'll always have the engineer unlocked to tier 5 after doing it once, if you don't perform missions for the engineer (or sell explo data to them, or take part in engineer related CG's or something), then your 'discount rep' drops continually to neutral over time. This then encourages you to bring them a few thousand in Exploration data, or specific commodities (like they ask for to unlock them, for example) when you visit to get yourself the material discount. Not only does this create gameplay and open avenues for new content if/when FDev choose to introduce a new Engineer CG/Mission/Task but it also increases the realism factor. If I were headed to Lei Cheung for a new Reinforced Shield upgrade in real life, I'd probably expect to take him some Gold in exchange for not having to bring him any 'improvised components' (You know they basically don't exist, unless you trade other G5 mats that were hard to get for them, right?), or if I was heading to Liz Ryder for an overcharged power plant upgrade, I'd probably expect to take her a box of 10 Kamitra Cigars so that I don't need to bring her Selenium (one of the hardest things to actually pick up in the galaxy) or Chemical Manipulators (Not actually hard to pick up or trade for), but still, a discount is a discount. You could also introduce an element of negotiation to this, where you bring Liz 20 cigars and ask her to give you the G3 engineering for free, no mats, or bring her 50 like you did to unlock her and you persuade her to do the G5 tier for free, etc.
Yes, there should be an abundance of G1 - 3 materials (I dare say even G4 tbh) at considerably more opportunities and make them easy to come by. They're supposed to be common, not rare. Blowing up a ship in a CZ should drop considerably more of the 'common' G1 - G4 manufactured mats than it currently does for example, as should dropping into an emissions signal. Whether that's done by dropping more 'items' or making each item worth more materials is probably neither here nor there, although sometimes collecting the materials as they float into space can be a challenge in itself, especially in a CZ or pirate threat signal where you're being shot at.
I feel increasing the availability of materials as mission rewards would go a long way to fixing this grind too, getting 3 MEF for completing a mission that took an hour, is like offering 1 'buy three, get one free' tickets to see the Ghostbusters reboot from a few years ago, at the cinema, because you sat through the Oceans 8 film... give 10 for completing a mission that took an hour, especially if it's a hard one! Maybe offer 3 if your reputation with the faction is poor and 10 - 15 if your reputation is allied, something like that.
Although manufactured mats are by far the most annoying to collect (Because they're so heavily used in engineering and so mind-numbingly repetitive to gather and drop in such small amounts), I would consider dishing out encoded mats more regularly too. I dare say, that scanning a ship (any ship) successfully should dish out encoded mats readily. The whole thing of going to Jamesons crash site and relogging is horrid, but it's less horrid than just sitting outside a station scanning ships repeatedly for a decade to upgrade one module...
As for the idea of engineers dishing out materials, I don't think this makes sense from a realism perspective. Mats are their price for performing the engineering upgrades, so perhaps a slight twist to this, which could be more realistic and more engaging, would be reputation with the engineer (Separate from the current reputation mechanic tied to engineering level), earning you a discount on materials. This reputation could drop over time as well, so although you'll always have the engineer unlocked to tier 5 after doing it once, if you don't perform missions for the engineer (or sell explo data to them, or take part in engineer related CG's or something), then your 'discount rep' drops continually to neutral over time. This then encourages you to bring them a few thousand in Exploration data, or specific commodities (like they ask for to unlock them, for example) when you visit to get yourself the material discount. Not only does this create gameplay and open avenues for new content if/when FDev choose to introduce a new Engineer CG/Mission/Task but it also increases the realism factor. If I were headed to Lei Cheung for a new Reinforced Shield upgrade in real life, I'd probably expect to take him some Gold in exchange for not having to bring him any 'improvised components' (You know they basically don't exist, unless you trade other G5 mats that were hard to get for them, right?), or if I was heading to Liz Ryder for an overcharged power plant upgrade, I'd probably expect to take her a box of 10 Kamitra Cigars so that I don't need to bring her Selenium (one of the hardest things to actually pick up in the galaxy) or Chemical Manipulators (Not actually hard to pick up or trade for), but still, a discount is a discount. You could also introduce an element of negotiation to this, where you bring Liz 20 cigars and ask her to give you the G3 engineering for free, no mats, or bring her 50 like you did to unlock her and you persuade her to do the G5 tier for free, etc.