"With 2 starting ships, I don't see myself ever attacking WSS in the opening rounds." And that's good, because tyrell is the most vulnerable house. Any other house has some means of defending: Stark can muster ships in shivering, has roose, bara has saalhador, martell has arianne, gj has the sword, lanni has the raven, but tyrell has to just pray for muster round 2, otherwise it's helpless against martell. It is absurd, let alone when you don't have a muster nor CoK for a few rounds.
When we talk about adding extra barrels, it's a good idea, this is just not the issue here. It's about the ships and losing the sea early. Also, tyrell for some reason doesn't have a strong sea card.
Tyrell's "protection" is that Martell has to traverse more sea space to threaten its core holdings, as opposed to Bara or Greyjoy's capitals, which can be completely isolated by one successful naval attack, or Lanni's capital being threatened at the start of turn 2, or Stark having to split its defenses across 3 areas on 2 sides of the map. That extra space gives it an extra turn to catch a Muster or a Clash, which also allows it to repel an invasion reliably.
But really you are just demonstrating my point. By attempting to "correct" for Tyrell's vulnerabilities in the early going, what you are doing is producing more stalemates. Which could pull the results further in the direction of perfect 16.7% balance (although that is not even sure, as any tiebreaker ruleset is going to favor some houses over others), but who care about the results if it creates plodding, lifeless games along the way? We need more vulnerabilities! Vulnerabilities create action! Action creates the chaos that makes games memorable!
Besides which, for all this focus on Tyrell's pitiable condition in the early game, how often do you actually see them overrun immediately? I would say it is at most 20% of games where I see Martell actually take WSS on turn 2, and less than half of those where they actually press that advantage into a full green wipeout. Stark is the house that I see getting wiped out first the majority of the time, with Lannister still probably being at more risk than Tyrell, even after the ship buffs for them. And yet, Stark's results are nearly the best, because if they do survive through the midgame, they wind up in a great position.
Which is where Tyrell is actually lacking, imo. Sure, their early game position could be improved, but it is more a lack of any clear endgame path to victory that keeps them from wearing the crown. Successful Tyrell games often look really chaotic because of this, with proper conquests of Dorne or Lannister being less frequent than some oddball assortment of Starfall/Harrenhal/CCP/Flint's Finger/Lannisport/King's Landing/Riverrun.
The supply buff is a (subtle) way to help with this, by making Tyrell more of a force to be reckoned as soon as they make it to the midgame.
As for the map changes, I see a few pointers here:
1. Reach: definitely an extra barrel
2. OldTown: I said it a few times, one of the biggest and richest cities, it should have a barrel and a crown. I can't see why it's "weaker" than Riverrun or Seagard.
3. The Arbor: might have a barrel as well. Whoa, too many barrels for tyrell? Exactly, that's one of the richest houses, able to field the biggest armies.
4. The Twins: should be a stronghold because it is one of the strongest castles in Westeros, though I'm not a proponent of adding this, as it would make things easier for gj and stark, which is exactly the opposite of the discussion.
5. I've always been wondering: Why is there no Casterly Rock? Why is Lannisport the capital? Those two should be placed next to each other.
This is mainly lore stuff, rather than gameplay. I don't know about buffing up Oldtown too much, as having two adjoining strongholds in such a protected position is already the core of Tyrell's strength. But I do like adding the Reach barrel for both lore and gameplay reasons, as it gives a little something more to contest or bargain over in the middle of the map, where 4 houses could show some legit interest in it, but generally nobody does because 1 muster point just isn't worth investing real effort into. Honestly, it's not even really a buff for Tyrell, since the upshot is that there will be more enemy focus on one of their "home" territories. But, more action!