Your DM is correct, 2024 Guidance doesn't work on initiative, because it's a Dex ability check that doesn't involve any skill proficiency. 2024 Guidance requires the caster to choose a skill.
The target adds 1d4 to all ability checks involving that skill for the duration. For example Charisma (Intimidation) when threatening to use political influence to make life difficult for an NPC, then Strength (Intimidation) when resorting to direct physical threat; these two different checks could both benefit from the same casting of Guidance.
There's no way to get the benefit of Guidance on raw ability checks where no skill proficiency could apply, like a pure Strength check to brute-force something if your DM doesn't typically let Athletics apply to lifting/pushing tasks. Or to counterspell/dispel checks with your spellcasting ability modifier.
Enhance Ability (2nd, concentration for 1 hour) does still apply to raw checks (like initiative, counterspell / dispel, etc.), since you still just choose an ability. You get advantage on all checks with that ability. (Not saves or attacks, just checks.)
2014 Guidance did/does work on initiative, but you had to already have cast it before the thing that causes you to roll initiative. It's a spell with 1-action cast time, 1 minute concentration, same as in 2024, not a reaction.
Casting it before opening every door is the power-gamer move (and narratively justified before doors in dungeons), if you're not worried about the noise from the verbal component and you're not already concentrating on something else.
The difference between 2014 and 2024 guidance is that 2014 adds 1d4 to any one ability check of the target's choice within the duration (https://www.dndbeyond.com/spells/2132-guidance), which could indeed be initiative, counterspell or dispel, or any other raw ability check. Then the spell ends after 1 use, unlike the 2024 version.
In 2024 the caster chooses a skill and all checks with it are buffed with +1d4. vs. in 2014 the target chose one check to use it on.
If you're not concentrating on anything else, the 2024 power-gamer move is still to cast Guidance on someone before opening a door, but you have to pick a skill that might be useful if there are enemies or a fast-moving situation. A decent default choice is athletics or acrobatics for escaping grapples, but note that the 2024 changed grapples so the person initiating a grapple doesn't make a check, it just imposes a save on the target1. Else perhaps perception (since that's useful for ambushes or for non-combat searching a room), or if anyone's combat abilities involve a certain skill (like insight for a rogue subclass) then that's an obvious choice. Or acrobatics at tables where the DM likes to call for that when PCs try to do cool stuff using terrain in combat.
If the party includes a rogue who repeatedly hides in combat for advantage, Stealth is an obvious choice, although many rogues already have high enough stealth that failing the check is very rare. (Especially once they have Reliable Talent.) Could be quite good at lower levels, though.
Guidance does have a verbal component so you can't do it silently without some other features, so casting it before opening a door could potentially alert creatures on the other side. But it lasts a minute so you could be some distance from the door.
Since the spell is concentration, the 2014 ending when used is detectable by the caster. This makes the game-mechanic of ability checks an observable feature of the world's reality for characters who inhabit it, not just players. This could either be interesting or really bad, depending on how you feel about characters doing basic research into how their world works, like some of the characters in Brandon Sanderson's cosmere books who research the magic systems of the different worlds his novels are set in.
The 2024 change patches that "hole", not making the exact moment of an ability check observable by anyone. (Silvery barbs has the same "problem", being a reaction to someone making a check, save, or attack, so implicitly assumes that characters can notice those things. It hasn't been reprinted for 2024.)
Early in the playtest leading up to 2024, Guidance was going to be a reaction to match how it got played at many tables out of initiative. (Someone calls out "guidance" after the DM calls for a check, and it's usually allowed if there would have been time for the narrative to include asking that PC to come over and guide them). Presumably they reverted it back to a full action and 1 min concentration because of the combat boost of making it a reaction, especially for counterspell. (e.g. wizard or sorc casts Counterspell, druid reacts to that with Guidance.) Would also have been OP for grappling if used with the 2014 grappling rules where it's a contested check.
Footnote 1: So Guidance isn't a bless-for-grapplers, only for escaping grapples. Like in 2014, some monsters have an auto-grapple built-in to their attacks, with an escape DC listed. Those monster auto-grapples use the same escape rules as standard grapples where it's a check against that DC. (In 2014, it was a contested check to keep a target grappled, like when initiating, except for monsters with a listed escape DC. In 2024 there's always an escape DC, never a contest.)
It's very rare for DMs to replace a monster attack with a grapple, so PCs will usually only find themselves needing to escape a grapple when fighting certain monsters. But there are enough of those that Athletics or Acrobatics might not be a bad choice for Guidance before entering an unknown situations. Especially with DMs that like to call for either of those checks for other stuff in combat like when trying to flip past enemies.
- Unarmed Strike - if you make your unarmed strike a grapple, the target has to succeed an Str or Dex save against a DC of 8 + grapplers Str + proficiency, which is also the DC to escape later. Same for Shove.
- Grappling - escaping a grapple is an action to make a Str (Athletics) or Dex (Acrobatics) check against the escape DC, instead of the save you make during the initial grapple.
If you were playing a 2014 game but had one PC with 2024 Guidance, casting it on a grappler PC would be very strong, giving them a +1d4 to every grapple they initiate. So probably not a good idea to do that. :P
