I don't think it's good. I mean in case you play in extremely low elo where every game is "80 kills / 30 min fiesta", then you have a lot of gold and can spend it for more offensive items like Titanic Hydra, Sunfire Cape, Abyssal Mask etc. to help yourself win the game by having more damage. On the other side, the classic supportive builds are much cheaper (= better when you don't get that much gold from kills) and in more coordinated games of "higher" elos your teams often have another tank, your teams don't int every single thing so your teammates profit more from items that help them instead of items that make you tankier and give you slightly more damage (but ofc this thing is connected to the less gold thing). You have to understand that - especially on higher elo - the benefits from Sunfire and Titanic Hydra are not that high; in current state of game, many champions are very mobile and people with hands don't come as 3-5 on one spot to allow you ult them and proc all these damage item passives you have on as many enemies as possible.

Yes, Leona is one of the strongest champions for lvl1 5v5s or 3v3s. If she manages to get in the center of the fight, the champion she stuns is easy to focus and kill. In case she starts with E, the engage potential is better, however the CC is shorter, damage output is lower and E recast time is much longer than in Q start case; E start is often a bait for Leona.
However, despite I'm aware of Leona's level 1 strength, I'm not the one that "wants all to fight level 1 as much as possible". You know - all level 1 actions are pretty flippy; often people sit afk base till 0:45 or something, some people refuse the invade calls, or the invades lead to 3v4, 4v5 or unfavorable, though numerously equal, fights. And the best part... the junglers who call invades, often fail 50-50 smites. So this all just means that I'm not that optimistic about the lvl1 chaos.

Sometimes exhaust can be good when you don't have much kill pressure on botlane and enemies have some hypercarries or assassins, but personally I take Ignite in like 98 % of games - which can partly be caused by my elo; high elo games are faster, more tend to snowball and ignite is better for these purposes - for 2v2 kills, roams etc. As this guide says, the most common general use of exhaust or heal on Leona is when Yasuo or Twitch on ADC role decides to pick ignite so that the adc has higher chances to get kills and Leona should pick a different summoner spell to help him instead of playing double ignite lane.
And about different supports than Leona, yes, exhaust is viable and pretty good (but not necessary - depends on matchup, ally adc summoner spells, playstyle etc.) on enchanters, against Tristana or Lucian, on defensive champions like Thresh, Braum, Taric etc.

Well, this is build for support Leona, I don't know anything about your games, but my experience is that games are pretty fast and - which is more important - supports don't get as much gold as in past due to rework of stacking support items and shutdown system. All these items usually used by toplane and jungle tanks are expensive, especially the Titanic Hydra. But yes, in low elo Titanic Hydra on support Leona is somehow rushable. You will have some tankiness, more damage (but it's not that amazing, especially because you have to autoattack to use that extra damage properly and sometimes you just can't get in range or you get stunned) and much lower utility for your teammates. And of course - you shouldn't get behind while rushing Titanic Hydra - in this case it's a snowball item.
Though, in my opinion, if you buy Titanic Hydra, Ingenious Hunter is not that good. It depends on a situation. If you decide to go for secondary Domination tree and you are in a situation where your engage is not based on your movement speed because you are permaslowed whenever enemies play it properly (for example against Morgana / Zyra / Senna / Janna / ...) and you wanna rush Titanic Hydra, yes, the Ingenious Hunter > Relentless Hunter. But mostly I think that the engage or roam potential provided by Relentless Hunter is stronger than CDR on an item like Titanic Hydra that is made for dealing much burst damage and then long time waiting for another burst. Often the fight is too short for you to use the active for a second time even if you have active item CDR. (And let's not forget some more options like secondary Inspiration or Precision tree that still can be better than Ingenious Hunter in those concrete games where Relentless Hunter is pretty bad.)
Generally, core builds that use Ingenious Hunter are these 2 (in situational order of building; Control wards or pots not mentioned):
1) T2 boots + support item + Redemption + Locket of the Iron Solari (+ might be Kindlegem)
2) T2 boots + support item + Righteous Glory + Locket of the Iron Solari
Both are currently slightly off meta, but imo more viable than that Titanic Hydra rush. In these builds, active item CDR is more effective because in both cases it increases amount of fights or catches you can take part in and without Ingenious Hunter you often miss a good possibility to use your items by few seconds.