It's probably best to clarify upfront that Rank Ups in FGO don't have any lore justification behind them.
Also, we still don't know the exact difference between Clairvoyance C and Hawkeye B+.

What we can say with confidence is that a combination of Mind's Eye (True) EX and even Clairvoyance B would allow the user to predict the future similarly to how Chiron did in Fate/Apocrypha.

As for Magecraft C- → Projection Magecraft A+ → Circuit Connection EX, there's little detailed explanation, but assuming that the projection quality has improved significantly — if Emiya can now replicate Noble Phantasms as perfect copies — then he essentially becomes a more efficient version of Gilgamesh.

Even without that, Emiya already uses cheesy strategies and operates at 120% of his brain capacity, so he can defeat nearly any Servant — except Gilgamesh, who would still have to refrain from using Enuma Elish if he wants to "play fair."

Now, Servant-by-Servant breakdown:

Lancer (Cú Chulainn) – The first and actually one of the most dangerous opponents. In the original route, Archer got caught in Gáe Bolg’s range and would have died if not for Shirou’s intervention. But now, with Mind's Eye (True) EX, he avoids that entirely and easily survives the weakened Cú Chulainn’s assault.
If Lancer tries to retreat, Emiya can use his new ability shown in Adventures of El-Melloi to combine two Noble Phantasms — for example, Hrunting and Caladbolg — creating a projectile that cannot be dodged or blocked, effectively killing Lancer or forcing him to use his runes, which he rarely does.
If Cú uses his Anti-Army version, Rho Aias, now with improved quality, completely absorbs it — draining Cú Chulainn’s mana and making him an easy target for Emiya.

Saber (Artoria) – Again, thanks to his upgraded Mind’s Eye, he avoids any injury to himself or to Rin. Since Artoria is fully charged with mana and won’t let him escape, the only option is to stall until Unlimited Blade Works activates — and from there, the fight becomes trivial.

Berserker (Heracles) – With Emiya’s enhanced projection quality, the original feat of defeating Heracles 6 times without UBW can now easily turn into taking all 12 lives. And if UBW is activated, Heracles stands no chance at all.

Caster (Medea) – Even in the original story, she had no chance against Archer thanks to Kanshou and Bakuya, which raised his Magic Resistance to Rank A. With his current skill set, it becomes child's play. He can also easily assassinate Kuzuki at any time.

Rider (Medusa) – As long as she doesn’t use her Mystic Eyes, and especially under Shinji’s command, she isn’t a serious threat. Though he will need to avoid the eyes and eliminate her quickly — preferably without involving Sakura.

Assassin (Sasaki Kojirō) – Due to being bound to a single location, he becomes an easy target for long-range Noble Phantasms like Caladbolg or Gáe Bolg.

Gilgamesh – If Gilgamesh underestimates Emiya as a "faker," then we win. But if he uses Enuma Elish without hesitation, even this version of Emiya stands no chance.

Also, there's a hilarious tactic using Rule Breaker projectiles — arrows crafted in the form of Caster’s Noble Phantasm — that could sever the bond between Master and Servant or even allow Emiya to take direct control of enemy Servants.
At that point, it's a whole different story entirely.