I'm going to start by saying this: This is not a system system, it's just the way I'm describing the ability/magic. I'm not going to suggest anything beyond the Class Cards, it will be a lot simpler that way. I'll be trying to suggest at least two Servants for each class, excluding Avenger, Moon Cancer, Shielder, Saver, Foreigner, Gunner, Faker, Gatekeeper, and Watcher. I may or may not include Ruler and Alter Ego. It all depends on what I choose to include, choice-wise. Let's start with the most popular class, Saber. I'm also getting most of my info from the wikia, so feel free to make any corrections.
So, I tried to be a little more original and didn't include Artoria in any of her various forms on this list, and I think I'm better off for it. The two Sabers I ended up choosing are Sigurd and Lancelot. I think Sigurd fits well because of his Dragon Kind Modification, which would give him a similar feeling to Dragon Slayers. His stats are: Strength A+, Endurance A+, Agility B, Mana C, Luck E, and Noble Phantasm A++. Lancelot would bring much more skill due to his Eternal Arms Mastership, allowing the MC the use of all weapon skills no matter what may influence his mental state. His stats are: Strength B, Endurance A, Agility A, Mana B, Luck E, and Noble Phantasm A++. His Skill A Knight Does Not Die With Empty Hands also allows him to turn anything he recognizes as a weapon into a Rank D Noble Phantasm.
My two choices for Lancer Class are Karna and Scáthach. I think both are powerful in their own respects, but if I personally had to choose one to use in a story, I'd pick Karna. While Scáthach definitely has more skill and is basically immortal, Karna has more appeal for this setting. Plus his Noble Phantasms are a much better fit, strong without being ridiculous. Karna's Stats are: Strength B, Endurance C, Agility A, Mana B, Luck D, Noble Phantasm EX. Scáthach's Stats are: Strength B, Endurance A, Agility A, Mana C, Luck D, Noble Phantasm A+.
I have to admit something, I did include an unoriginal choice: Gilgamesh. I mean, I also chose Atlanta, but I know more people will complain about my first pick. At the very least, I didn't suggest EMIYA. Gilgamesh seems to be the easy pick, especially with the Gate of Babylon, which would be perfect for all manner of functions. Atlanta is the more standard kind of Archer, with one unique twist. She has a second Noble Phantasm which basically acts as an A-Rank Mad Enhancement, upping all her parameters except Luck by one Rank. I think it would be annoying to keep writing the Parameters down, so I'm going to stop.
It is time for Riders, who will be Francis Drake and Ozymandias. Francis Drake has a fleet of ships, which can be called at any time. Oz basically says no to other Servants' Noble Phantasms. Get the picture? Great, moving on.
Caster's are Solomon and Tamono-no-Mae. Solomon for pure OP and Tamono for uniqueness. Assassins are Shiki Ryougi and Shuten-douji, Shiki for her Mystic Eyes and Shuten-douji for poisons. Finally, we get to the Berserkers, which are Minamoto no Raikou and Beowulf. Personally, I lean more towards Beowulf, due to his various Noble Phantasms, two of which can actually break, to my surprise. His third doesn't come into play without the destruction of the first two, which is a really unique since I haven't seen any other NP break before, unless you count the minor ones that Gilgamesh uses, which I don't. I was going to also include Ibaraki-douji, since it would be cool to have her sync with Shuten, but I finally decided on Beowulf and Minamoto. Sorry I didn't give these classes the same dedication as the first three, I realized that if I did it would cause the chapter to drag on.
There is a novel with a similar premise taking place in DC, which has a neat idea, the cards would have a slight ego, being able to dictate when they are used. They also, after a long enough period will imprint all their memories onto the MC. While a neat concept, I'd rather our MC continue on being himself, but the ego's will come into play for a later idea. I would like to suggest the possibility of Ascension, which causes the Heroic Spirits to grow closer to their prime as a form of growth for the ability.
Now, the concept I had in mind was that this magic would be similar to Requip and Celestial Magic. The Requip part is due to the ability to install the cards onto the protagonist, which causes him to gain the equipment, Parameters and Skills of any of his Servants. The Celestial part would come from an additional power, which would allow the MC to summon any of his Servants, as long as they aren't currently installed. This would come at the cost of increased Mana depletion and as long as they are summoned, they cannot be equipped.
Obviously, this was meant as a reincarnation story, but I could see transmigration as well.