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.

Added.

"Start with red trinket. Expect Shaco boxes in river bushes after you pull for jungler or in lanebushes. Watch Shaco's manabar to see whether he casted Boxes into areas without your vision. Don't chase Shaco or his adc without red trinket or perfect knowledge of position of his boxes. Don't E on Shaco who is pretty close to his turret - he can use Q under turret, becomes unvisible and you dash under enemy turret and take turret shots. Remember that Shaco has only 1 relevant CC spell - boxes, so often you can kill his adc if your adc is not low hp (-> won't be killed by Shaco)."