i think the answer is because of eagle man and lion of the golden city, their cost will be reduced for each neutral card u play, that means you can just put them in a really low cost, so i think it will basically somewhat having "aggro" characteristic
if whether lion of the golden city can be played without amulets in hand, well, it depends on the situation, best situation is to play it along with amulet, so you will gain strong advantage in early or mid game, because with lion of the golden city, u able to make ur board full at once...
i think it's not that necessary bring sacred plea or temple of the inquisition, because usually when u play neutral haven, ur board will be full at once thanks to the lion of the golden city cost being reduced by all neutral followers you play, dont forget to put goblin princess in ur neutral haven deck, because it will be reduce either lion of the golden city or eagle man by 3 in one go
The main premise of Neutral Haven is self-explanatory : You bring the neutral package to overwhelm your opponent with buffed neutral minions while simultaneously using Haven cards to somewhat control the board and refill on threats using a cost-reduced lion that insta-pops amulets. It's just a fancy way of saying "Aggro Haven". Think of it as a variation of Storm Haven that relies more on board presence instead of blowing your opponent out of the water using storm minions.
You shouldn't care about lategame. If you want to care about lategame, play Aegis Haven. There's a reason why it doesn't run Scripture. There's a reason why it doesn't run Tribunal. There's a reason why it doesn't run Themis. And the reason is the most basic concept of aggro decks : Face is the Place.
TL;DR : Sacred Plea is too slow for a deck that relies on board presence, likewise with Temple of Inquisition. You'd rather play minions instead. I'd even argue Wise Merman is better than Sacred Plea.