Any witch can do in combat healing with the Life Boost hex taken via Basic Lesson feat at 2nd level (lesson of life).
Primal and Arcane witches do control best, everyone but Arcane does healing and condition removal with spell slots.
Solid Fog and Wall of Force are still there.
I would recommend the Silence in Snow Patron. That will give you the Clinging Ice Hex cantrip that will likely qualify as a snowball vending machine.
It also gives the Primal spell list that has Heal spell, though you will not have an ability to swap in healing spontaneously.
As Xenocrat mentioned, Basic Lesson as a level 2 feat will give you the Life Boost hex which is a healing over time ability. It may not be the best emergency healing during combat though.
If more spontaneous casting is desired, there is the Flexible Spellcasting class archetype that Witch does qualify to take. That would make all of your spell preparation spontaneously cast - similar to the PF1 arcanist. It does cost overall spell slots though, so be aware of that.
For control spells, the Primal list has spells like Mud Pit, Ash Cloud, Mist, Wall of Water, Solid Fog, and Wall of Stone.
Wall of Force is still available, but it is not on the Primal list.
I think you might be misunderstanding what the remaster is. It is not a new edition, like PF1 to PF2. You can still use all the old PF2 content.
The remaster is simply an errata pass that is doing the added work of rebranding and tweaking things just enough to not get sued. Web is not likely to be coming in Player Core 2, because its function was covered by a new spell: Entangling Flora. If two spells different spells did the same thing but for different classes, they got combined into one. That not only rebrands the IP, but saves page space and reduces the data a player has to sift through. So instead of wizards casting webs while druids cast entangle, both now have Entangling Flora. Other examples:
Light + Dancing Lights = Remastered Light
Fly + Air Walk = Remastered Fly
Glitterdust + Faerie Fire = Remastered Revealing Light.
Wall of Force is still called Wall of Force for some reason, but Magic Missile is now Force Barrage. I don't think Solid Fog was ever in PF2, and in general fog spells are much weaker than they were in PF1. (Actually, lots of spells are weaker, so temper your expectations.)
Page 188 of Player Core has a sample build called Hedge Witch, but I'm not sure it does what you want. The PF2 witch can never spontaneously cast spells. They are prepared. But they do have a lot of healing options still:
Cauldron: Lets you make a free potion each day. Can be healing potion, though by mid levels there's much more creative things you can make.
Basic Lesson: Lesson of Life. Grants you Life Boost, a focus spell you can basically use for infinite out of combat healing. (As long as you can spare 10 more minutes.) Alternatively, take the blessed one archetype for Lay on Hands, which is also infinite out of combat healing and better for in combat.
Beyond that, it just comes down to Patron and spell selection. To maximize your stated goals, you'll want either the Resentment or Silence in Snow.
Silence in Snow
Your hex cantrip inflicts a speed penalty and your familiar creates difficult terrain, which is about as good as infinite, level 1 battlefield control goes. You are a primal caster, which means you have heal (which is just bonkers powerful compared to the PF1 cure spells), Entangling Flora, Wall of Stone, Toxic Cloud, and most of the spells which literally reshape the battlefield. That said, you could also go...
The Resentment
Good old Evil Eye is back, as well as the PF1 style cackle of extending debuffs forever. Resentment uses the occult list, which means you get the weaker heal instead of soothe. You also can't reshape physical matter as much as a primal caster, so no walk of stone. That said... The best battlefield control spells (especially at low levels) are actually illusory object and illusory creature. And keeping a target slow for the rest of its life even when it succeeded on its saving throw... I guess you could call it a debuff instead of battlefield control, but they certainly won't be moving around the battlefield as well.
I also promise I did search for the spells I mentioned and did not find them. Not sure what happened there. Maybe a glitch as AoN updates things? I do software myself and I'm willing to blame goblins hiding in 4U servers :)
It took me several minutes to find Ash Cloud in the list of Primal spells even though I knew both the name of the spell and that a Druid ally of mine was casting the spell when we were level 3 characters.
I have no better explanation of what caused that for me than the goblins you proposed.
Primal is the healing/blasting tradition. It's also good for snowball vending.
If the thing you want is healing and buff/debuff, then you want the Occult list. Personally, I think that sounds more "battlefield control" than primal, though Primal does have summons and (eventually) walls.
protip: Summons aren't amazing in PF2. There are ways to make them worth the spell slot, but it takes knowing when and how to use the things, and it's not anything like OP even then.
The only tradition that can't heal is Arcane.
Apologies if I came across as condescending. The gaps left from the aforementioned goblins were peculiar and there has been a lot of misinformation around the remaster.
I will reiterate that occult has really, really good battlefield control. Illusory Object and Illusory Creature are my picks for the best two spells in the game. They are also still amazing cast in low level slots, so picking them up through an archetype is viable if you pick a primal caster.
Calm would also be the best incapacitation spell in the game, depending on how your GM runs it. But there's enough table variance there that I'd suggest talking to them before you pick it.
With proper spacing, Illusory Object can easily rob a solo enemy of 2 actions with no save. (You'll need to go just before the enemy to avoid hurting your allies too, though.) Illusory Creature can similarly bait a brute enemy into wasting multiple actions, and give your front line martial a flanking bud, trigger weakness, teleport around the battle field, and attack from absurd distance. My last session, my 4th level occult witch dealt 60 damage to a troll with one illusory fire elemental, and that doesn't count missed paladin strikes the flanking turned into hits. Later, I finished off a badly injured dire wolf with an illusory archer, then baited a second into attacking it while my party and I focus fired on the third and fourth wolves.
Occult has more limited wall and cloud options. But walls don't get very good until high levels and cloud spells are just less value now that they only provide basic 20% mischance regardless of distance. These illusion spells kick ass out the gate.