The paradox is fixed once you understand the 3 iterations of fate are different time loops. The first (The fate storyline) Is the first pass thought. It follows 4 rules.
1. The well of heros exists outside of time, so heros past present and future can be summoned.
(This allows Archer to be summoned at all)
2. Using the Gem as her summoning tool was the Paradox/Window heroic spirit Emiya used to get into the war.
(This is the catalyst to start a potential time paradox)
3. The grail itself is sentient, and knows all details of the servants it summons. It does not allow paradox.
(This is a counter force the grail has to combat paradox)
4. Time loops in fate are not paradoxes if they are not infinite.
(This allows Gaia and Co (all the celestials of the Nasu verse to make multiple attempts to salvage die situations)
So in the fate time line when Archer is summoned, that is the original start point for the loop. This is why Archer in the first loop has trouble remembering his own past. The grail erased it to stop him causing a paradox. Its only when he is defeated by berserker with the utterance "And i thought i had thrown them all away." Where he recovers his memory and feels regrets (the things he thought he had thrown away. For this arc his destiny is undecided, so right up until his final moments, when the grail is sure that the timeline of shiro becoming a heroic spirit and the paradox is fixed it returns Archers memories to him.
In this way Archer for the first arc represents the Idea of the hero Emiya, but doesn't have his will or motives.

This leads to a Shiro who was more powerful in the second loop, Creating the "Unlimited Blade works" Arc.
Because of repetition,  the loop is now established there is no paradox. Archer is now already and forever in the well of heros. In this arc the archer we meet is the completetion of the story of the Shiro from the first loop. So it gets to have motive will and goals. This Archer is a fully realized version of Shiros Broken Idealism. He however, faced with his younger self, and seeing that Shiro gets a major power up just by fighting him give him hope that it will work out better this time. The next Shiro will be stronger and know his dark fate. He may never have to make the Guardians pact. Accepting this Fate arc Emiya hands the baton the UBW Emiya and sacrifices himself.

This leads into the third arc, "heavens feel". Where the Archer we meet is the final version of the UBW Emiya. Its why he is a bit more chi