There's a big difference between 3rd parties and Nintendo characters though. As long as they're important characters to their series, any Nintendo character should have the chance to get in. Marth, Ice Climbers, Shulk, Wii Fit Trainer.
However, I think it's best that 3rd parties span the generations when it comes to their franchises, like Sonic, Mega Man, Pac-Man, Rayman, Crash Bandicoot, etc. Those characters are ALWAYS the leads in their franchise. Simon Belmont only really represents the NES era of Castlevania. Snake has represented MGS from the beginning to present day. You could rename MGS "Solid Snake" and it would still work, but there have been too many Castlevania protagonists for it to work the same for that series.
The red part is factually incorrect. Solid Snake is not always the protagonist in Metal Gear games. Note the green ones are the games that are actually on Nintendo systems.
CANON
Metal Gear - Solid Snake
Metal Gear 2 - Solid Snake
Metal Gear Solid - Solid Snake
Metal Gear Solid 2 - Raiden
Metal Gear Solid 3 - Naked Snake
Metal Gear Solid 4 - Solid Snake
Peace Walker - Naked Snake
Ground Zeroes - Naked Snake
The Phantom Pain - Naked Snake
NON-CANON
Snake's Revenge - Solid Snake
Ghost Babel - Solid Snake
Acid - Solid Snake
Acid 2 - clone of Solid Snake
Portable Ops - Naked Snake
Metal Gear Solid Mobile - Solid Snake
Metal Gear Solid Online - none but Solid Snake shows up as DLC
Metal Gear Touch - Solid Snake
Metal Gear Rising - Raiden
All this proves that Solid Snake is the most recurring protagonist of Metal Gear games and apparently the most recurring protagonist of Metal Gear games that actually showed up on Nintendo games. So, your argument about renaming the series "Solid Snake" would not work.
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Now, for the Castlevania games... we have had many more games that were on Nintendo systems... and despite the fact we have had more than a dozen different protagonists in Castlevania games, Simon Belmont was the most-recurring protagonist compared to them.
ORIGINAL SERIES
Castlevania (NES) - Simon Belmont
Simon's Quest - Simon Belmont
Haunted Castle - Simon Belmont
The Adventure - Christopher Belmont
Dracula's Curse - Trevor Belmont & Alucard
Super Castlevania IV - Simon Belmont
Belmont's Revenge - Christopher Belmont
Castlevania Chronicles - Simon Belmont
Rondo of Blood - Richter Belmont
Bloodlines - John Morris & Eric Lecarde
Symphony of the Night - Alucard
Castlevania Legends - Sonia Belmont
Castlevania (N64) - Reinhardt Schneider & Carrie Fernandez
Legacy of Darkness - Cornell, Reinhardt, Carrie & Henry
Circle of the Moon - Nathan Graves
Harmony of Dissonance - Juste Belmont
Aria of Sorrow - Soma Cruz
Lament of Innocence - Leon Belmont
Dawn of Sorrow - Soma Cruz
Curse of Darkness - Hector & Trevor Belmont (unlockable)
Portrait of Ruin - Jonathan Morris & Charlotte Aulin
Order of Shadows - Desmond Belmont
Order of Ecclesia - Shanoa
The Adventure ReBirth - Christopher Belmont
LORDS OF SHADOW
Lords of Shadow - Gabriel Belmont
Mirror of Fate - Trevor Belmont, Simon Belmont & Alucard
Lords of Shadow 2 - Gabriel Belmont
SPIN-OFFS
Kid Dracula (NES) - Alucard
Kid Dracula (GB) - Alucard
Judgment - Simon Belmont, Alucard, etc.
Harmony of Despair - Alucard, Soma Cruz, Jonathan Morris, Shanoa, Charlotte Aulin, Simon Belmont (DLC)
Let's tally the notable recurring protagonists... keep in mind I am not counting spin-offs.
Simon Belmont - 6 games (4 on Nintendo systems)
Alucard - 3 games (2 on Nintendo systems)
Trevor Belmont - 3 games (2 on Nintendo systems)
Soma Cruz - 2 games (2 on Nintendo systems)
See? Simon Belmont is the most-recurring protagonist of Castlevania... and the most-recurring protagonist in Castlevania games that are on Nintendo systems. Do you know what is a plurality? This is an example of one... yes, the majority of games feature different protagonists but Simon Belmont is the most-recurring protagonist compared to all of them. This is why I consider Simon Belmont the Marth of Castlevania series... it started with Simon Belmont and he's the most-recurring protagonist despite not appearing in many of them... just like how Fire Emblem started with Marth and he's the most-recurring Lord despite not appearing in many of them.
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Oh, and by the way... the Pac-Man games didn't always star Pac-Man... sometimes they star Ms. Pac-Man... hell, even Mega Man and Sonic weren't always the protagonists in their respective franchises... so no, using the "the character is always the protagonist of his/her franchise" argument is not a good argument against third-party characters. In those cases, it is better to go with the iconic/important/most-recurring ones.