Serious question, why even attempt to fix the jungler's when the itemization is so messed up? Right now there is nothing you can actually do to the tanks that will put them on par with anyone that can use Warrior's enchant UNLESS you make them stupidly overpowered, in which case they will never be played in the jungle and they will simply crush lanes (Maokai anyone?).
Normally I give Riot the benefit of the doubt but the new jungle enchantments are TERRIBLE, so I am going to post this giant wall of text hoping some of these issues are addressed in the changes.
Short version: AP Champions deal approximate 30% less damage in the jungle compared to their AD counter parts, this discrepancy occurs entirely due to auto attacks. Tanks are in an even worse spot. That aside all enchantment build paths are complete shit compared to warrior enchant due to partial component cost.
This may be easiest to explain looking at how individual stats interact in the jungle and what getting more of this stat provides for a jungler.
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AD. If you get more you kill things faster. The faster you kill things the less time you spend fighting monsters, less in combat time means less damage taken PLUS fewer abilities used meaning less mana used. Meaning by adding AD you are actually also sorta adding health and mana. This is easily demonstrated by playing the same champion and buying a long sword (and no potions) and buying a ruby crystal. Then going and clearing, at the end of the same number of camps the champion that bought the long sword will have the same or higher finishing health AND mana, PLUS they will still have the 10 damage the long sword provided them. So in the end the champion that bought the ruby crystal is out damage, health, and mana.
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AP. In large quantities AP is good and allows you to clear quickly, however it typically only applies to abilities resulting in weak auto attacks. As an example at level 1 while an AD champion gets slightly more AS and about 20 extra AD from runes and masteries, the AP champion will get about 30 AP. In lane this would be roughly even. This isn't the case in the jungle since AP does not meaningfully improve auto attacks at early levels. Meaning even if there is a 1.0 AP ratio compared to a .5 AD ratio on a first ability. The AP champion will get 30 damage from AP then wait 5-10 seconds to use it again. The AD champion on the other hand will get 10 damage from the ability PLUS 20 damage onto every single auto attack they make. In 5-10 seconds you are looking at 4-8 auto attacks (80-160 damage), that the AD champions pick up just in the first spell rotation alone. In an initial clear of 3 camps, AD champions will be about 750 damage ahead of their AP counter parts when they finish, or to put that in more contextual game terms roughly 10 auto attacks plus a round of abilities for the AP jungler.
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Health. More health means you can take more damage. In the jungle flat health is really actually pretty bad early it doesn't give you any damage so you end up taking more damage than what any health itemization or runes/masteries give you. At least armor reduces the damage you take improving the efficiency of health pots, health on the other hand does not. For this reason many tank junglers end up with damage oriented runes and often take some damage related masteries to allow them to make it through early clears. The other problem with health is as jungle camps level up it still provides no damage, the time for a tank jungler to clear a camp remains high through the entire game. While any type of damage jungler will see faster camp clears as they build items.
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AS, more autos, more damage, more damage, less combat time, less combat time less damage taken and fewer abilities used, less damage taken and fewer abilities used more health more mana. It is a solid stat.
So you can already see some pretty clear issues with the stats that SHOULD have been addressed in development of the item, and I would say as completed items go Warrior, Magus, and Devourer are all approximately close to each other, although Magus does lag behind slightly (a different story we are talking about tanks), Juggernaut on the other hand is pretty terrible, as an item it is okay except it is needed in the jungle. There is a reason why even J4 who is strong has almost exclusively swapped over to Warrior over Juggernaut, if he goes juggernaut he will rapidly begin to fall behind, and we are seeing the same issue the old jungle items had, that the early damage allowed you to tank up faster.
Build path: I am glad to hear you are looking at build paths because right now Warrior's build path is 10x better than any other build path, because longswords are OP.
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Long sword vs ruby crystal buy: Junglers back with 400g, AD jungler gets a longsword and a health potion. As a result they get 10 damage and 150 health for their next journey into the jungle. The tank gets 150 health. Guess who does better? Surprise the long sword wins, not only do you kill camps faster taking less damage and using less mana, you can get that health back, this allows you to apply pressure in lanes or at worse clear another camp that will more than pay for that health potion, all in the same amount of time. The sad thing is it gets worse for other items.
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Long sword vs amp tome: Junglers back with 435g, the AP jungler walks out with 20 AP. The AD jungler has choices now! They can walk out with 10 AD and 2 pots (300 health) or 10 AD and a ward. Plus it is still early in the game where auto attacks are a major portion of the damage in early jungle clears. So not only does the AD jungler still clear faster, they either have more consumables than their counter parts or they have a ward to get vision advantage.
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Long sword vs dagger: Junglers back with 450g. AS jungler gets a dagger, the AD jungler has choices upon choices. They can wait a second or two and get 3 potions, a green ward and a potion, or a pink ward. Again all around massive advantages that in this case will easily translate into more benefits for the AD jungler compared to their counterpart.
At the end of the day AD junglers have an advantage at level 1 because AD is simply the best stat in the game for clearing a jungle (well rivaled by AS at level 1), simply because auto attacks are such a major portion of the damage in the early jungle. While building the enchantment AD junglers once again have an advantage simply due to the cost of the longsword being significantly cheaper than other enchantment components, even if it sets the jungler back 35-100g on a back that money is more than made up with the potions/wards that are bought. Normally you trade off a strong early game for a weaker late game, AD junglers don't actually lose an advantages later in the game.
So what each jungle item needs: AP item needs sustained damage (give the machete burn a 0.7 AP ratio). Tanks need their stats to provide some form of improved clear, not only early in the game but it needs to scale later in the game. Sapping barbs that were introduced was a good concept but lacked the scaling into late game that the damage builds had. AS junglers are close enough to fine they are a pretty specialized group and with infinite scaling comes balance issues.
I also just want to say with the addition of PBE tracking jungle statistics. This will show the discrepancy between the stats, but it will not show the discrepancy caused from the advantageous build path for warrior enchant. One thing I hope you collect data on (not show obviously) is when different jungle enchants are completed. The cost of all three four (whoops) enchants are the same, but what is the average completion time for the various jungle enchants.
All around I also just want to say thank you for finally beginning to address the persistent issues that are in the jungle, this was not a problem caused by the jungle changes, these issues were prevalent even in the old jungle and it is largely itemization driven due to the challenge of balancing an item for multiple champions.