Not helping is the dependence on accessories that forces Summoners to forego movement-enabling accessories in favor of just a tiny bit more damage or survivability
Which Stage of the game is Summoner Accessory dependant? Because from what all I can see based on my summoner playthrough experience and guides there's enough accessories for summoner exclusive accessories and movement accessories. Also 10% damage and 1 Minion slot isn't even tiny, it's a large damage boost for minions.
Obsidian Armor being viable in Hardmode being a very clear example of that
Obsidian is not Viable in Hardmode. It got hit by multiple nerfs and spider armour is better for pre-mech) A better example of Summoner progression issues is pre-boss and pre-mech. Pre-Boss requires a hell of a grind and Pre-Mech barely has any options only having one (Or Two) options for Qs to get blade staff. (Note: I find it really funny that Summoner headgear has like 1-3 defense. Not complaining tho)
While I like your idea of Summoner headgears, I do think there's should be a balance of Whip Armour and Summon based Armour. Being that 1.1 Ores are Summon Based Armour (Their Set being Raw Damage) While 1.2 Ores are While Based Armour (Their Set Bonus complements Whips)
Heres My Suggestion:
Cobalt Crown: 2 Defense (22 Defense Full Set)
12% Summon Damage
15% Whip Attack Speed
1 Minion Slot
Set Bonus: Whiplash
Allows another Whip to be switched out after the whip the player equipped attacks 3 times. Bound a Whip for the passive to be usable.
Example: Using Cool Whip then attacking an enemy 3 times switches the Whip to Firecracker
(Since Whip Stacking is here is stay make it easier or make players learn about Whip Stacking, and I gave it more speed and defense since it's a whip based armour)
Palladium Hat: 1 Defense (19 Defense Full Set)
17% Summon Damage
2 Minion Slot
Set Bonus: Rapid Healing
(Hard Time balancing this because of Spider Armour, but I made this a summon based armour because of the healing passive, so Spider Armour should all be about summon Damage Whip Palladium Armour should be all about healing)
Mythril Mask: 1 Defense (21 Defense Full Set)
15% Summon Damage (7% extra damage from chainmail)
2 Minion Slot
Set Bonus:
20% Increased Summon Critical Strike Chance
(Since Summoner Minions doesn't actually crit, (except for whip tag damage) I want an armor that can give Crit Chance Pre-Mech without the need of Whip Tag)
Orichalcum Crown: 3 Defense (26 Defense Full Set)
9% Summon Damage (8% Extra Damage from Leggings)
2 Minion Slot
15% Whip Attack Speed
30% Whip Attack Range
Set Bonus:
Flower Petals will fall to your target for extra danage
(Flower Petals buff whips)
Adamantite Hat: 2 Defense (30 Defense Full Set)
18% Summon Damage (8% Extra Damage from Breastplate)
2 Minion Slot
Set Bonus:
Increased 20% Minion Attack Speed
1 Minion Slot
(I don't know what do to with this other than this armor having the most Summon Damage)
Titanium Hat: 4 Defense (30 Defense Full Set)
12% Summon Damage (7% Extra Damage from Leggings and Breastplate
2 Minion Slot
20% Whip Attack Speed
35% Whip Attack Range
Set Bonus: Titanium Barrier
Chlorophyte Glasses: 5 Defense (2Se36 Defense Full Set)
17% Summom Damage (5% Extra Damage from Platemail)
2 Minion Slot
30% Whip Attack Speed
15% Whip Attack Range
Set Bonus: Leaf Crystal
And do you still have the sprites? Can I use them? I want to make an idea suggestion for other hm ore headgears