I think the nerfs make his laning less oppressive, but not awful. He'll still be strong, just not in as good of a place as before. The mana nerf doesn't do much as you go Ancient Coin often, so you don't miss out on mana regen. The hook being increased by 4 seconds at level 1 does mean you may not have the liberty to put a point or two into Flay in most games, as you need to get the cooldown shortened ASAP.

Hextech Flashtraption is useful on champs that lack the range to actually get in and do their job without making it obvious. Alistar is someone who uses it well, because the range of his Headbutt is relatively short. Since your Death Sentence and Flay are both ranged abilities, you aren't at a range disadvantage. While it's not necessarily bad on Thresh, there are better options that aren't as niche as Hexflash.

It's basically Ancient Coin only as of late.

At this point in time, Relic Shield should almost never be considered. The health is worse, your auto attacks are ranged thus not providing the full healing benefit and making acquiring stacks harder, and the bonus gold generation is sub par. Ancient Coin is providing move speed for roams, a good chunk of CDR quite early in the game, and great gold regen., as well as the mana coins.

The build you are recommending is more based on utility. However, for the sake of efficiency, Ruby Sightstone has good synergy with all the actives you have, giving you 20% CDR to all of them (which isn't achievable through normal CDR).

My play style tends to be a tanky initiator who jumps right in rather than a utility based support who hangs back. If I'm not teamfighting, I'm playing a Warden role where I'm defending for my carry, getting picks or peeling when we're outnumbered.

In terms of Face of the Mountain's shield capabilities, it's great for immobile champs or squishy casters. If your carry or mid-laner happens to be much more susceptible to assassins (people like Jhin for example), it's going to save their life. Assassins who do more damage the more health the enemy is missing (like Zed) are especially weak in that regard) as their initial burst is absorbed by the shield and they won't be able to do that extra damage.

In the guide I do state Eye of the Equinox should always be considered, but it needs to be in the right situation. If the team possesses an assassin or person who can do high single target damage, it can be a lifesaver (especially in conjunction with Locket of the Iron Solari). However, if their comp favors more sustained teamfights where there's no risk of someone getting 100-0'd, a Eye of the Equinox with a Redemption would be more favorable.
The build is always subject to change based on the enemy composition, but I've found this works well with how I play Thresh.