There's no guarantee you will have a Soul Shard available. You can try plan ahead by always saving one but that will most of the times result in a DPS loss. Since it's completely random when the trinket procs you're forced to play reactive and ignore the possibility of a proc occuring until it happens. I don't have the trinket myself but I'm sure a situation people end up in many times is to having to choose between UA/Corr after refreshing Agony or just use Fel Flame to get all dots buffed (many times with a short duration because Fel Flame only gives 6s).
No, UVLS is better for afflic than Hydra since you can put 3 dots up easy in the 4 second window, and maybe even a haunt if you're fast enough.
That's not realistic. Pooling 3 soulshards in the hope of such a rare trinket proc will be a dps loss, regardless of the 100% crit. Wushoolay's + Breath of the Hydra is a better combo than either + UVLS.
UVLS is undeniably, unmistakably the most powerful trinket this tier for both aff and demo. That is, if you can use it properly, else it will indeed be shit for you. But that comes down to player - the trinket itself is ridiculous. That will be for every individual warlock to decide but there's honestly no discussion possible on how OP this trinket is for dotclasses, especially warlocks, but also spriests and moonkins.
With that much haste and the legendary meta, having no shards is, well, something of the past. Pooling shards for really good procs is a dps gain, not a loss, as long as you don't cap on your shards. If you don't have a shard when UVLS procs then somewhere along the way you made a mistake in your dps planning, and anyway, it's perfectly possible to hardcast 3 dots in 4 seconds. If I don't have a shard when I get 3 amazing procs at the same time I simply did it wrong, means I wasted shards on inferior procs or worse, on no procs at all. It's the same for UVLS.
Anyway, as OP gathered, going full haste is simply the way to go now due to the nature of trinkets and the meta.