Yes, Plane Shift can bring you to any layer of the 9 hells

Evidence from the Plane Shift spell

The spell Plane Shift specifically says that it can transport the party directly to the palace of Dispater, which is located in Dis, the second layer of Hell. Thus it can clearly bring you directly to the lower layers of Hell.

Getting the material components to enable this transition, however, will be difficult. To utilise the spell you require a tuning fork specifically attuned to the layer of Hell you want to plane shift to.

Finally, Plane Shift doesn't create a portal, it transports the people affected by the spell to a different plane of existence.

Contrast this with the 9th level Gate spell which specifically states:

You conjure a portal [...]

Even if Plane Shift did create a portal, the fact the spell text specifically says it can transport creatures to Layers other than the 1st Layer of Hell would be a perfect example of the specific beats general rule.

Evidence from the DMG

The DMG section on the Outer Planes has two big caveats that we can drive our caravans through down to the lower levels of the Nine Hells:

1. It states (emphasis mine):

   Planar portals don't connect to the lower levels of the Nine Hells by Asmodeus' orders

   • If planar portals couldn't connect to the lower layers, there would be no reason for Asmodeus to make this order. As a result we infer two things:
     1. Asmodeus has told the other lords that they are not to allow the creation of planar portals on their layers (aside from Avernus).
     2. Planar portals can be connected to the lower layers...they just aren't created by the native denziens, who would likely work very hard to permanently close/guard any that did exist due to the aforementioned orders.

2. The Infinite Staircase section states

   [...] on any given plane, there can be multiple doors to the Infinite Staircase, though entrances aren't common knowledge and are occasionally guarded by devas, sphinxes, yugoloths, and other powerful monsters

   • This is certainly one type of portal (using the literal meaning of the word portal to mean door) that connects different planes, which is explicitly designed to connect to every plane of existence.

The DMG also has a specific section on the Plane Shift spell (under Planar Travel), discussing it's limitations.

It seems balanced, but...

At first glance it seems balanced. It does not have anything particular worrisome or that it may break the class or abuse the system in any way. But, when I looked at the patron a bit closer I found some glaring problems.

The easiest one is the first two abilities "Nature's Aid" and "Earth Friend". Usually, when designing a class or subclass, you have to be sure that a feature should be usable from the level that it was gained. If not, the penalty that was in place seems unnecessary. In this case, this part is the one I'm talking about, emphasise mine:

Nature's Aid: ... Alternatively, you may use your bonus action to empower a minion - any celestial, fey, fiend, elemental or beast that you command, such as a familiar or summon

As you can see, for this feature to be useful, it needs a minion. Unless I'm not aware of another warlock spell, the lowest summoning spell that a Warlock could get is "Find Familiar" with the "Pact of the Chain" boon, followed by "Summon lesser Demon" (a 3rd level spell). If the warlock chooses not to get any summoning options like those previously mentioned, that section punishes the player. This is where "Earth Friend" might help. This is just one of many options that you could use to improve the situation. That is, it is just an advice. Change "Earth Friend" for something similar to:

Earth Friend:

At 1st level, you gain the ability to cast "Find Familiar" as a ritual. If later you gain the boon "Pact of the Chain" you can change "Find Familiar" for one ritual spell from the Druid's spell list of level one and cast it as ritual.

Moreover, if at any moment you choose the "Beast Speech" Eldritch Invocation, you can speak with beasts and plants telepathically when using that feature.

Why the change? First, it make it possible for the player to fully utilise "Nature's Aid" since now the player has the chance to have a minion. The next part was put there to not diminish in power the "Pact of the Chain" boon. Swapping for a similar level ritual spell do not seem to have unintended consequences, at least for now. The last part is also to keep your idea kind of intact AND the old "Earth Friend" feature seemed a bit redundant given that "Beast Speech"'s Eldritch Invocation exist.

As an example for this type of changes, the "Hexblade" patron has a clause similar to this, emphasis mine.

Hex Warrior: If you later gain the Pact of the Blade feature, this benefit extends to every pact weapon you conjure with that feature, no matter the weapon's type

And this include and also extends the Eldritch Pact "Improved Pact Weapon".

Improved Pact Weapon: Finally, the weapon you conjure can be a shortbow, longbow, light crossbow, or heavy crossbow.

Wall of Elements

Considering the most damaging Patron feature "Hurl Through Hell" does a 10d10 psychic damage with no save and make the target disappear for one turn (given that an attack hits), some of the damage options of the Walls are a bit over the top. A creative use of the wall can, potentially, make one or more creatures useless for more than one turn, making it a better option in comparison to other Patron features of the same level. However, it is a spell, can be counter-spelled, and limited by concentration. Furthermore, the other options of this Patron, are on the weak side (2.5 of damage on average per attack per 3 turns is not that powerful). I'd just lower the walls to 6th level and call it a day.

Cure wounds

The second biggest problem I found is that it does not have cure wounds. Now, it might not seem like a problem becase there exist other options: Multiclass and feats. But, those are optional rules. Thus, I evaluate the lack of this considering these facts. I understand that you are probably safe, but I find that not adding this section won't make my answer complete.

"Nature's Aid" uses a pool of 1d4 heals to heal a target. The limit on the number of 1d4 that you can use are set by the Charisma modifier. Given a +5 modifier, at worst it heals 5 HP and, at best, 20 (with an average of 12.5 HP). A bit worse than the Celestial counterpart (minimum: 5, maximum: 30 and average: 17.5). Because of the limit given by the charisma modifier, it scale terribly in combat at higher levels. "Cure wounds" scale better in combat (Using a 5 level spell slot, "Cure wounds" minimum heals for 10 HP, a maximum of 45 and an average of 27.5, with a +5 modifier).

As a (possible) main healer, loosing this might hurt in the long run. And if you want a more Druid alternative, "Healing spirit" is a really good one (a 2nd level spell). It is better than cure wounds, but adding arguably on of the best healing spells might consider too powerful for a Patron spell.

Conclusions

For the most part, it is balanced, in particular considering other Patrons. The only major issue I saw was the "Nature's Aid" problem and a bit of redundancy with "Earth Friend". The walls might be a problem as a free cast, but there are ways to deal with them and the other features balance the walls nicely.

The "Cure Wounds" problem is only bad if you consider that Multiclass and feats are optional rules, and that you might be the main healer. Aside from this, it is a solid Patron.