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How would one build an emperors children list?

I like Palatine blades, Phoenix guard, and would like to include Fulgrim

I don't like Kakophani

A pride of the legion list with two veteran tactical units and lord commander eidolon is one of the cheapest starts you can go with - anything else can be added from there and also isn't a massive outlay if you decide against PotL lists in the future while also being useful

The crux of an EC force is getting the units to where and when you want them, so high mobility units are super useful and your special stuff should have a metal bawkx or something to get to the party in style.
Maybe a Kharibdis Assault Claw? no reason, i just like seeing them in action.
Support elements like Javelin attack speeders are worth a look as are the classic land's speeders - standard javelins are 75pts with a TL cyclone launcher and a HB; the classic speeders are 75 with a graviton gun and multi-melta upgrade so they can add anti infantry or anti tank for not much outlay which i think may be important at the points level you are considering.

I generally advise people avoid using their Primarchs at 1,000pts just starting out; you tend to end up with a list that has the Primarch, some minimum mandatory units, and nothing else.

The Emperor's Children rules push a melee-centric force, but almost all their bonuses are written around making sure you get the charge and break your opponent on doing so. Phoenix power spears give you masses of AP2 at a higher Initiative than anyone short of Custodians so long as you can charge, but they're just power swords if you don't charge.

Palatine Blades and Phoenix Terminators are both specialist elite-infantry-killers; they don't get enough attacks to fight hordes, they don't have the Strength to take vehicles/monsters, and they'll lose to things that have the Invulnerable saves to tank the first hits. In a thousand-point game you'll probably be using one unit, and I'd suggest jump-pack Palatine Blades since they're a lot cheaper than delivering a Phoenix squad.

Rites of War are an interesting question for the Emperor's Children; my usual advice is to take Pride of the Legion and use mechanized Tactical Veterans as Troops, but in the Emperor's Children you want units that can get themselves to melee quickly (bikes, jetbikes, jump infantry). There's no Rite that puts Outriders in Troops, and Sky Hunter Phalanx doesn't let you use Jump Infantry in the army at all, so it might be worth doing Angel's Wrath or Drop Assault Vanguard.