UPDATE: Adventurers League changed the "+1 rule" in 2021.
An announcement from Q1 2021 gives the details. Basically, Adventurers League players may now simultaneously use any and all of the following sources when choosing player options like races, classes, subclasses, feats, spells, and backgrounds:
Other source books may also be available depending on the specifics of the campaign. The currently active campaigns (as of June 7, 2021) draw from subsets of the following list:
Note that variants and optional rules must be explicitly allowed, so the variant Human, Half-Elf, and Tiefling races in these books may or may not be available. The same goes for the customized origins in Tasha's. The best thing to do is to find and follow the character creation guidelines in the Adventurers League Player's Guide specific to your campaign, which can be found here.
The revised rules have been implemented in these active guides:
However, the active Seasonal Campaign Player's Guide (Plague of Ancients) remains the version from September 15, 2020. It's likely to be updated when the next Seasonal Campaign begins.
The active Guidelines for Using TCoE in Adventurers League Play also predate the change, so must be revised.
Goliaths
Regarding the original question, Volo's Guide to Monsters is on the list of source books available for all Adventurers League campaigns. Because goliaths are described therein, they're allowed in Adventurers League play.
However, under the updated rules, choosing a goliath no longer prevents the use of character options from other allowed source books like Xanathar's Guide to Everything.
"Appropriate level" means one with which the spell could normally be cast
The relevant quote from the new Fey Touched and Shadow Touched feats is (TCoE, p. 79-80):
You can also cast these spells using spell slots you have of the appropriate level.
In this case, the phrase "spell slots [...] of the appropriate level" means you can cast the spells using any slots that would normally be appropriate for you to cast the spells with. There is no ambiguity here, as "spell slots" is plural modifying the subject to which the "of the appropriate level" applies to to be the totality of the spell slots you have.
So you can use the base spell slot level of the spell, or you can opt to use a higher-level spell slot as outlined in the PHB rules:
When a character casts a spell, he or she expends a slot of that spell's level or higher, effectively "filling" a slot with the spell.
So, at a minimum you have to use a slot of the spell's level, but are free to use a higher one.
Is the word "the" not intended to restrict it to only using a single slot level (the base level)?
If the feature was intended to only allow the spells to be cast at the base level, the wording would need to be "You can also cast these spells using a spell slot you have at the base level of the spell." or "You can cast these spells using appropriate spell slots, but only at their base level."
What about the Artificer Initiate feat?
One of the other points raised is that in another feat in the same book, Artificer Initiate (p. 79), the corresponding part of the description says:
You can also cast the spell using any spell slots you have.
Here, it is unambiguous that you can cast using higher-level spell slots. So does this mean the text for the Fey Touched and Shadow Touched feats was intentional to prevent being able to use "any slots"?
There is a significant difference between the Artificer Initiate feat and the Fey/Shadow Touched feats. The Artificer Initiate feat only allows you to learn 1st level spells, whereas the Fey/Shadow Touched feats allow for 2nd level spells as well (e.g. misty step). Given that 5e is explicitly an exceptions-based game (via the "specific beats general" principle), if the same wording were used on the Fey Touched and Shadow Touched feats, it would create an exception allowing the 2nd-level misty step to be cast using a 1st-level slot (given that 1st-level is a valid choice from "any" spell slots, but not from the "appropriate" spell slots).
It's also worth considering, that not all feats (or indeed features) are written by the same writer, and that 5e is written using natural language, as opposed to strictly defined and interpreted language. There are multiple ways to convey the same meaning using natural language. As a result, trying to put two features side-by-side and using one feature to try disprove a feature application in the other due to wording differences won't be appropriate in most cases.
Best Answer
No, gift of alacrity is not AL legal.
The first page of the DDAL Player's Guide has a list of books from which you may select various features, the sidebar "WHAT RULEBOOKS SHOULD I USE?", and Explorer's Guide to Wildemount is not on that list:
When selecting spells for an Adventurer’s League caster character, your complete spell list is actually the union of several different lists: the base spell list for your class from the Player’s Handbook, the expanded spell list for your class printed in Xanathar’s Guide to Everything (lists start on pg. 147), the expanded spell lists offered as optional class features in Tasha’s Cauldron of Everything, and the draconic magic spells printed in Fizban’s Treasury of Dragons. I’m pretty sure all the spells that were first printed in Sword Coast Adventurer’s Guide and the Elemental Evil Player’s Companion were reprinted in Tasha’s Cauldron, though if I am mistaken the options printed there are available as well (as noted in the quote, you must use the most recent printing of a spell in AL).