An exhausted creature can benefit from a bonus to its speed.

A condition does only what it says. The grappled and restrained conditions, as you cited them, state explicitly that the creature can't benefit from any bonus to its speed, but the exhaustion condition states no such thing, nor is there a general rule prohibiting the benefit of bonuses to speed when a creature's speed is reduced to 0.

Specific beats general, so a spell like longstrider would supersede the more general exhaustion condition (on the assumption that one spell out of hundreds is more specific than one common condition out of about a dozen). Otherwise, if that were not the intended precedence, then the grappled and restrained conditions would not even need to state the exception.

This is upheld by Jeremy Crawford, lead rules designer for 5e: "Having 5 levels of exhaustion reduces your speed to 0, but it doesn't prevent bonuses to speed from applying to you." (Source)

This is not surprising if you consider each condition thematically: grappled and restrained represent external factors locking a creature in place to restrict their ability to move (and gaining a bonus to speed would not circumvent that restriction), whereas exhaustion represents the creature's fatigue robbing them of the energy or gumption to move (but gaining a bonus to speed, likely via magic, could overcome that by effectively energizing or motivating them).

The Elementalist Fighter class is more powerful at every level than a standard Battlemaster Fighter.

I'm using the Battlemaster Archetype as a reference point as the OP has stated that it is an influence on the homebrew class design and because the class features closely mirror that of the Battlemaster.

TLDR; Elementalist Fighter blends almost all of the features of a Battlemaster, adds a sprinkle of Barbarian and then adds a tonne of powerful abilities to boot.

• Second Wind comes 2 levels later, decreasing survivability at 1st and 2nd level, before returning to the status quo at 3rd.
• Weapon Bond is almost identical to the Eldritch Knight feature and comes 2 levels early. It somewhat undermines one of the EK's unique features but is not too major.
• Imbue Weapon is a reskinned Barbarian Rage, the quintessential Barbarian feature. It is more offensive focused than defensive. There doesn't appear to be an end condition, but I'll assume it's the same as Rage. Almost everything here feels out of flavour for an elemental focused fighter. Psychic resistance, temporary darkvision, changing the advantage on checks and saves to Charisma, and advantage on saves against being charmed are not, in my opinion, things I would associate with a class called "Elementalist Fighter". Mechanically, choosing the additional damage's type is strong as it somewhat enables low-level melee fighters to circumvent physical damage resistances, one of their greatest banes. It is somewhat comparable to the 3rd level Elemental Weapon spell, minus the to hit bonus and magical weapon consideration. Losing the physical damage resistance of Rage is not a suitable drawback, as Fighters are much more capable of achieving high ACs at low levels than Barbarians, through heavy armour, shields, and fighting styles. So the loss of defensive abilities is negated by the classes intrinsic high AC and the damage capabilities are greater than the standard Rage, with some odd extras thrown in.
• Maneuvers. Having extra restrictions on maneuvers is sure to only cause confusion. The save DC being Charisma based is also quite unexpected and again outside of the classes flavour. Some of the new maneuvers are far above the current power level. Elemental Slam stands out as particularly egregious. Fighters lack AoE as a balance requirement. Giving them access to AoE which does good damage for a relatively low resource cost, is definitely going to cause them to be overpowered relative to another fighter of the same level. In conjunction with the powerful secondary abilities that many of the damage type Slams offer, I don't know why you wouldn't pick this maneuver over any other.
• Ancient Blade doesn't have any real balance concerns. It does, however, cause a great deal more work for your DM, which is something that I would consider poor design. Know your Enemy of the Battlemaster is pretty whatever, but it doesn't cause any additional preparation or forethought that could be used on the game as a whole, instead of one character.
• Elemental Incarnation is absolutely insane. Indomitable is already a strong class feature, and EI blows it out of the water. It evaluates to casting one of the Investitures of X spells, without the need for concentration, once per day, at the cost of one level of exhaustion. And this is in a non-casting class, 2 levels earlier than a full caster could. The sheer damage output and/or utility that this feature could provide in its 10 minutes is incredible. Especially considering most conditions that require a save also give one each turn. Potentially missing a turn or two is well worth this.
• Elemental Explosion is again powerful because it gives Fighters access to powerful AoE at a relatively low cost.

If given the choice to play a Battlemaster or an Elementalist Fighter, the Elementalist is the clear winner. With its access to cheap AoE and additional damage output, it does everything a Battlemaster does and more. It suffers from a lack of identity, borrowing iconic features from other classes, and also having a mix of features that don't resound with the classes core concept.