There are several issues regarding breeding mechanics. I'll discuss each one with possible solutions.
Issue #1- Male and Genderless Pokémon's incapability to gain Egg Moves:
This is a pretty important issue that plagues the generic Male exclusive and Genderless exclusive Pokémon.
Male and Genderless Pokémon cannot have Egg Moves, which disables their ability to gain older moves and Move Tutor. To balance things out, they are often given large TM movepools and Level-Up movepools. Unfortunately, this doesn't help a lot, as the Pokémon's older moves still end up lost in the sea of generational shifts.
This is also a major issue when it comes to gender counterparts, as more often than not the female counterparts become superior to their Male counterparts. This is the case for Miltank and Vullaby.
Issue #2-Alternate Forms and Hidden Abilities:
This is another issue that has recently emerged. While we have successfully got rid of the whole event exclusive HAs, this stays as an issue. Because certain HA Pokémon are only available in specific forms, and because forms are effectively considered as different species, it makes it difficult, if not impossible to obtain certain HAs with certain Forms. For example, White Flower Flabébé can't be found with Symbiosis. While this might be to done to make things "easier" competitively(read, you basically telegraph your HA with your specific form, see East Sea Gastrodon in X/Y for example), it just comes off as incompetent at worst, and frustrating at best.
In addition to above, there are a few miscellaneous issues;
Limited Egg Group Bridges: Much like new type combinations, each generation introduces at least one new bridge between Egg Groups. However, these are often limited and have limited movepools that render them useless as a bridge. Some gimmick Pokémon like Farfetch'd and Spinda has large movesets, but overall that is not enough, even if they are useful.
Grandpa clause: This is only an issue for Nidorina and Nidoqueen. Since they were made incapable of breeding, they have been useless in breeding. Though Female Nidoran is adequate, it forces the player to keep Nidoran unevolved to ensure its breeding capabilities.
Female Egg Move transfer: This is not an issue that needs to be fixed, but rather a perception that needs to be fixed. Though Female Pokémon can grant Egg Moves, they are incapable of transferring their own moves, such as Lovely Kiss to Smoochum. This will likely be fixed in future games, hopefully starting from the next Kalos games.
Other issues such as limited Egg Moves, TM Egg Moves and Incenses/Baby Pokémon have been discussed in the previous posts, so I won't delve deep into them.
How do we fix these?
Proposal #1, More babies and retcons:
In order to fix issue one, the easy solution would be to provide gender neutral baby stages for certain duos, such as Tauros and Miltank. With this way, Tauros would be able to gain Egg Moves, while Miltank would get to keep unique Egg Moves. This is not an excellent or complete solution, but it is solution that benefits both lines each others' moves.
For other male only species, such as Throh and Sawk, stat neutral babies in the mold of Tyrogue an be provided. This can also be done for other counterpart Pokémon, such as Plusle/Minun, Lunatone/Solrock etc.
However, Vullaby and Rufflet are already baby-like in most areas except their in-game mechanics, which makes the idea of a convergent baby ridiculous. For these two, the only solution would be to retcon the two as gender counterparts to each other, similar to Volbeat/Illumise and Nidoran-lines, or provide each other another gender counterpart, like a real female Braviary and a real male Mandibuzz.
However, if there were to be a species with no other counterpart, then any idea of a baby would fall flat.
For genderless Pokémon, their mechanics can be altered in two manner;
Manner #1: Asexual reproduction: This way, the species would be able to gain special Egg Moves that can be inherited without changing the gender mechanics. However, this would require its own set of mechanics, such as held item triggering Egg Moves, to provide a wide selection of Egg Moves.
Manner #2: Mock genders. This would provide each genderless Pokémon a "gender" that is only relevant to interspecies breeding. For example, a "male" Magnezone would transfer genetic information and material with a "female" Magnezone, but they would be incapable of breeding with other species. This can be expanded into sub Egg Groups, to provide something similar to bacterial genetic transfer or computer to computer data transfer. For example, Bronzor and Claydol would be able to breed, but Bronzor would not be able to breed with Cryogonal. This would be far too complex however, despite being less complicated.
Manner #3: Item triggered Egg Moves. This would keep the current mechanics intact, but the items of each Pokémon would trigger Egg Moves when mating with a Ditto. This doesn't fix Ditto's incapability to breed, however.
Proposal #2: Dual parent species inheritance:
This would be much more simpler than the previous mechanic, but also much more complicated. With this, each parent would have a say in the product baby. For example the union of a Murkrow and Vullaby would provide both Murkrow and Vullaby. This would mean that Male exclusive species would be able to inherit Female parents' and Male parents' moves, while Female exclusive species will inherit both as well.
This provides an issue with the gender distribution, however. Eevee and Fossil Pokémon, for example, are created to be Male biased to avoid easy breeding. In order to keep this bias, a bias towards Female member can be presented. Alternatively, each species could have a unique inheritance value that would determine exactly how many Eevee or Smeargle would be created, for example.
This would also provide an issue with determining the inherited parent, but Everstone can be upgraded to protect the genetic line of the holder parent.
This doesn't fix genderless Pokémon's incapability to inherit Egg Moves, however. As such, one of the former solutions should still be used.
Miscellaneous issues proposal:
To fix the above issues, best solution would be to grant existing Pokémon additional Egg Groups. For example, Kirlia can be brought into both Amorphous and Human-Like, while Gothita can be put into both Human-Like and Fairy. This wouldn't work for all species such as Tauros, but it would work for most of them.