Flight from Edinburgh to Paris-airport arrival time

What airline? Each airlines's desk opens at a different time.

https://www.edinburghairport.com/inside-edi/for-passengers/check-out-your-check-in-desk-opening-times-at-edinburgh-airport

If you're flying at 6am then you must be on Virgin Atlantic. Four hours before (when their check in supposedly opens) feels excessive to me. I would say between 3am and 4am for arrival at the airport. 4 hours before is probably for their TA flights. It's hard to imagine them opening at 2am for flights at 6am to both Amsterdam and Paris.
If going by cab I would order it for 3am, or by bus on the 3am bus. Even the bus only takes 22 minutes at that time of the morning.

I would want to be there by four. Not sure what relevance your nationality is.

Virgin Atlantic doesn't fly between Edinburgh and Paris. I'm guessing it's Air France. Two hours before is fine.

That flight is an Air France airbus according to Flight Radar, but it is a code share with 9 other operators including Delta, Singapore and Virgin Atlantic
This morning it was 28 minutes late departing, although it arrived right time at CDG.

That you are American is irrelevant; it’s an international flight to an EU country for ALL passengers. I would arrive no later than 4 AM.

No more than two hours.

If you have no bags to check, and can get your boarding pass the day before, you head straight to security. If you have bags to check, the airline desk doesn't open until 4. Why get there earlier?

Yesterday, I flew from London to Budapest. My flight was at 7 AM. I arrived at the terminal just after 5 AM. I was in the lounge drinking coffee by 5:30.

If you look on the Edinburgh Airport home page at any time you can see the time it takes to get through security- https://www.edinburghairport.com/

Thank you everyone for the helpful information.