That's a lot of damage indeed. But haunting everything in sight + swarm sounds very bothersome. (I do need to cast BOTH for it to work, right?) I don't particularly care for dots [or pets]. Even in D2 I played bone necromancer and ignored all pets and dots. Bone spear, ghosts and corpse explosion all day, every day.
Basically, it's 125% damage boost + very easy to upkeep and not bother, vs having to cast 2 particular spells on all targets just to get the damage boost.
Currently trying to theorycraft my gear / skills. I hope it plays out well enough. If I struggle to hard I can always just swap. For the moment, I just try to maximize my build without having to resort to tactics I personally find unfun (like a pet dedicated build).
Like, what would be the best gems for a straight forward grasp / firewall / tornado build?
Bane of the Trapped is a given. It seems very great. Gem of Efficacious Toxin sounds ridiculous, bit I can't judge that since I haven't played. Is it any good?
For the last I'm unsure. Is Esoteric Alteration a good defense gem? How much can you feel it? Molten Wildebeest’s Gizzard sounds nice as well as def option. Lastly. Bane of the Stricken sounds neat, but I guess it's more for fast-attacking characters. Does it procc with Efficacious Toxins?
Spamming Locust Swarm is indeed bothersome, because it is a huge mana drain to cast it even once. To get around this, a lot of folks will prefer put the legendary staff The Wormwood into their Kanai's Cube, which will put an aura around you that will cast Locust Swarm automatically for free once every few seconds. Since it spreads on its own from there, you can get this effect for free through the staff effect rather than actually put Locust Swarm on your bar. That just leaves you to apply Haunt, which while being a single target (or double target depending on your rune choice) per cast, is a pretty cheap spell and can be spammed somewhat freely, and jumps to new targets when the current target dies.
If you're trying to ignore pets on the Witch Doctor, I'd consider looking at the Jade Harvester set; I know you mentioned you don't care for maintaining Soul Harvest, but the truth is the Witch Doctor has become highly dependent on the buff (which is a lot easier to maintain now, by the way) due to the Lakumba's Ornament bracer. Since that bracer now grants 6% damage mitigation per stack of Soul Harvest, once combined with the double stack cap from Soul Harvester ceremonial knife and the Armor increasing rune of Soul Harvest itself, trying to run without it leaves you with an unbelievably huge hole to try and solve once you start getting into things like Torment X or higher, and especially even higher Greater Rifts. Aside from Soul Harvest though, the Jade Harvest set has nothing to do with pets and lets you spread way more powerful DoT effects and then drop enormous nukes. Because of the combo of Jade Harvester set buffs and the Ring of Emptiness change I mentioned before, in 2.4.2 the set got a lot more powerful, with the only caveat being that it is still significantly behind the Zunimassa or Helltooth based pet builds in terms of damage mitigation.
In general though, the way I feel like is most effective to play Diablo when starting either a new ladder / season or just starting out altogether is to let the drops become your build plan, and not go in with a set build in mind. When you're starting with no options, the best thing you can do is let the drops you do get in the early game lead you to the skills you should be using. That will allow you to level the fastest, and be the most efficient right as you are hitting 70 and can get your first Haedrig's Gift set pieces, which in turn you can continue to allow to guide how you should build. It's really a bit later once you are in a position where you have accumulated enough items and power where you can farm effectively around Torment X difficulty that you might want to consider actively trying to target farm other pieces and move your build in a different direction. That way you don't set out with an idea in mind, have no luck getting the drops that will enable your build to really be effective, but be stuck trying to use your build idea in low difficulties and lowering how much loot you are getting. Thus you wind up progressing towards your actual goal dramatically slower and without anything to really show for it.
As far as offensive gems, you're right that Bane of the Trapped is generally the best offensive gem. Witch Doctor in particular has little trouble finding access to slow effects from skills, and the damage bonus is one of the largest multiplicative bonuses that applies to all enemy targets and all your damage. So that should have a spot in your gear. Beyond that, since you're not using pets and won't need Enforcer, you'll likely want something like Bane of the Powerful at the outset which is another good "all-around" legendary gem that gives you another +20% multiplicative damage (which doesn't scale up per rank, instead the duration you get to keep the buff after killing an elite pack goes up) and then even more damage versus Elites. For the third slot, it'd depend if you need the defense or can afford to run all three offensive; something like Bane of the Stricken is good if you are running a build that struggles a bit with single target DPS (and thus will need to spend more time fighting a Rift Guardian), or if you need the defense Esoteric Alteration tends to be the most effective defensive option in terms of raw mitigation. Gem of Effacacious Toxin tends to be more of a group play support gem, where you apply the poison so your entire team gets the benefit of the increased damage from all sources, as the damage of the gem's DoT effect isn't that impressive being spread over 10 seconds.
I think the legendary gems are actually a really good target to work on first, alongside doing a couple rounds of bounties and unlocking your Kanai's Cube when you start the new season so that you can more quickly get the big boosts from the legendary gem effects and get your first three Kanai's Cube passive effects active as soon as possible. That will be a lot more effective than trying to focus on a particular skill build while you're still at the mercy of your early game drops, since those legendary gems will be useful in pretty much any build. For your initial Kanai's Cube conversions, I'd probably look for either items that will benefit most builds (something like the Convention of Elements ring or The Furnace 2H mace which will offer damage boosts to any build), or key pieces that will compliment your initial Haedrig's Gift set to allow you to jump further up in difficulty sooner (say a Tasker and Theo gloves, Mask of Jeram voodoo mask, or Ring of Emptiness). If possible try to convert legendaries for Kanai's Cube that have a lot of variance in their legendary effect, rather than something like say Short Man's Finger which always confers the same benefit from its legendary effect without any roll range. Items that are converted into Kanai's Cube grant their legendary effects at their maximum values, so it's a good way to turn what might happen to be a really crappy rolled legendary into something much more useful.
. That reminds me, as Rogg said, you can PM me your Battletag and I'll add you to the EU clan, if you're not in there.