Can someone help me decide between an amp and an integrated amp?

There is an inherent problem in having separates: money! No component you can name will sound very good just sitting there on its factory supplied footers with its factory supplied power cord and its factory supplied patch cord. That's just a fact. You need to upgrade the power cord, interconnect, and footers with cones. At the very least. Then there's the fuse, to name just one of several very well advised tweaks that can make a cheaper component sound much better than its much more expensive competitor.

Pick any component, at any price, connected to those freebie wires, its very easy to demonstrate you can beat it handily with the same amount of money in a cheaper component with better wire. And cones. No contest.

This holds true at every level and for every component. So it really comes down to, do you really like spending a lot more money for less total performance? Or do you really need whatever little extra flexibility you might get with separates?

That's the usual advantage of separates, flexibility. Only in this case while you would seem to be gaining flexibility in reality you're tying your hands, severely limiting your choices.

You're looking to upgrade to Maggies. Okay. Whatever. Everything said above applies here as well. I decided long ago to not consider difficult to drive speakers, not because they might not sound great, but because of exactly these kinds of constraints they force on everything else. Years later when you inevitably decide to change and find yourself with a bunch of stuff that was chosen mostly because the speakers limited your choices, years from now well just maybe you will understand why I made the choice to avoid difficult to drive speakers. 

But hey, these are all your choices to make. You are well advised to leave my opinion, and everyone else's, out of the equation. All I'm saying is go into it with your eyes open.

Its elitist to acknowledge the fact that you can do better spending less by paying attention to details like power cords? Getting more for your money is elitist?

What seems to have you so worked up elizabeth is this:

No component you can name will sound very good just sitting there on its factory supplied footers with its factory supplied power cord and its factory supplied patch cord. That’s just a fact.

Well, that is in fact, a fact. Which in spite of what you seem to think is nowhere near the same as saying you are required to buy the power cord, etc. What it says is what it says. I find it helpful to try and read and understand what people actually say and write, and not stuff my made up prejudice into their mouths. YMMV.

@bradkava,
If you want to save space and possibly some money buy a good integrated amp with enough power to drive Maggies. With an integrated amp, you don't need to purchase interconnects for a preamp.
If you have space and funds then individual components are an option.
It's a matter of personal choice and no one but you knows what your preference is. I also should mention that I believe in the long run it's worth waiting in order to purchase components that are really good vs mediocre.
For example, this would do the job nicely: https://www.audiogon.com/listings/lis9die5-luxman-505ux-mk-1-integrated-amp-solid-state
I'm sorry to say I don't know if using your headphone amp would work as a preamp. I think we would need to know the specifics of the headphone amp. It would also be helpful to know a dollar amount you are willing to spend.

Hi not sure which Maggie's you intend to purchase and have no experience with the amps you mentioned but I did use a Parasound integrated with 1.7i and the sound was amazing.
I guess separates or integiverated just depends on the best available deal. Just be sure you have enough current to drive Maggies.

That Classe is a really nice piece for the money. 

Maggies, huh? Put Bryston on that list, be surprised how many maggie owners prefer Bryston. Should be able to find something used close to your budget, but a few more $$$.

Well my man, you have opened a can of worms. Here is my two cents, in general, separates sound better given similar build quality. Different power supplies running pre and amp make a difference. I prefer dual mono amps to a single amp for better separation and imaging. As was mentioned, you have flexibility with separates to mix and match for desired sound. Down side is at least one more power cord and one more set of interconnects with separates. While you don't have to spend a fortune on ICs and PCs, substantial gains can be made by upgrading over stock. Rack space is also a consideration. There are some very nice integrated amps out there but I am partial to separates. For what it is worth, I think what Classe CA series, McCormack DNA series, Bryston SST and Belles (what I still own) are tremendous amp bargains in the used equipment arena. All these are pretty much bullet proof. I also love tube power amps, especially well executed EL34s but I am tired of hassling with power tubes and the cost of nice NOS tubes is getting redunkulous as well as needing more power for my 4ohm 86dbB speakers. All of life is a compromise, so now I am happily running my Belles 350A Reference amp with a Mapletree 6NS7 tube preamp. With your Maggies, SS is the way to go, unless you rob Fort Knox and buy some bemouth tube monos. I would personally start with an amp with 200wpc into 8 ohms which doubles into 4 ohms. If you want a Luxman integrated, go the 507 route and not the 505 because the 507 will double into 4 ohms (even though I think you need more power than these for the Maggies) plus it is just a better piece of equipment.  I ramble and thus digress, I like a good SS power amp for control and authority (and worry free operation) and a nice tube pre amp for warmth and dimensionality. You could probably do very well with a couple of well selected previously owned pieces in the $2-3K range for both a quality SS power amp and a tubed pre. My 2 cents, YMMV.

Anyone who thinks something like an amp doesn't need a good power cord, interconnect and speaker cable to sound good, could you please tell me: How many audio salon's have you gone into where the power amp and pre amp was plugged into the stock factory power cord? With the factory patch cord? And lamp cord going to the speakers?

Has anyone been to ANY stereo store ANYWHERE and seen this?

Anyone? Bueller???

How many audio salon's have you gone into where the power amp and pre amp was plugged into the stock factory power cord?

My gods!! This proves absolutely nothing and I cannot, I say again, cannot believe I haven't thought of nothing before.