Most "special attacks" in 5e either just replace standard weapon attacks or are "on hit" effects that you can decide upon after making a weapon attack and hitting (e.g. divine smite, maneuvers, sneak attack). You should be able to do these on each of your attacks unless the ability specifically specifies that they're a once per turn ability (sneak attack).
As for Larian's homebrewed special attacks, a lot of these are way too powerful to let players use them more than once per combat, let alone once per turn. Special weapon attacks should only be a once per rest thing and a lot of the more powerful ones need to be removed. Some of them are fine due to being situationally useful and not that much more powerful than regular attacks, such as cleave which lets you once per rest attack up to 3 enemies in a small aoe with damage equal to a regular weapon attack if the attack rolls are successful. Others are extremely overpowered, especially those that inflict some of Larian's homebrewed conditions.
The barbarian special weapon attacks are a unique thing that definitely need to be examined. Anything that inflicts a condition should require a saving throw and should be situational such that they don't replace normal attacks such as diving strike's prerequisites. Anything that hits more than one target or deals more damage than a normal weapon attack needs to have sort of restriction/resource (maybe once per rage or an amount equal to proficiency per rest). They also really need to take a look at a lot of the persistent damage conditions and effect since they really don't work with a lot of 5e's mechanics such as concentration.
As an aside, Larian really needs to add a penalty to Frenzy since getting an extra weapon attack that deals full damage at level 3 is pretty powerful. I get that in tabletop the exhaustion penalty is so damaging that berserker is just worse than other barbarian subclasses, but there needs to be atleast some downside to getting a second attack before level 5. Perhaps they can give a temporary level of exhaustion that lasts for a single round during which you can't rage again.
Larian also needs to change the way a lot of on-hit "special attacks" work so that they can be used together. In 5e tabletop you can do something like a battlemaster maneuver and a smite or a reckless attack and a sneak attack in the same attack provided that the conditions for each are met. You can't in BG3 since these are considered separate types of actions/attacks which is going to be a real problem once multiclassing or certain feats are introduced.
tldr: homebrewed special attacks should have some sort of restriction or resource unless they're specific enough that they will not replace normal attacks. Anything that's directly from 5e should work the way it currently does in 5e. Special attacks should not make normal attacks redundant or uncommon. You should have reasons to just hit somebody normally.
