What about characters who gain no extra attacks nor cantrips?
You mean Rogues? Their damage scaling mostly comes in the form of Sneak Attack, so maybe Sneak Attack dice?
Also, if all my cantrips don't scale, do I get my sidekick for free?
All damage dealing cantrips scale, so if you haven’t taken any damage dealing cantrips, I wouldn’t exactly describe that as “free.”
The main problem here is that extra attacks and cantrips are not comparable with each other, and they are not comparable with sidekicks either.
Extra attacks and cantrip scaling are absolutely comparable, they fill the exact same role in the system math, multiplying the character’s average damage per round at specific level intervals. Are Sidekicks comparable to that? In some ways. Two attacks from a fighter is certainly comparable to one attack from a fighter and one attack from her warrior Sidekick. One 2d8 cantrip is comparable to two 1d8 cantrips. Obviously Sidekicks can do more than just damage, so in other ways that isn’t comparable, but I also think that’s kind of the point. If you want to run a game where the PCs attract followers as class features, you want them to do more than damage. Replacing Extra Attack and cantrip scaling is just to keep the party’s expected damage output in line in such a game.
In 3e there was a core feat, although in the DMG, called "Leadership", which granted you a cohort (plus various followers) of a lower level. If a DM wanted to give sidekicks as a character option (not mandatory), my first idea would be to consider it cost a feat, because at least a feat has the same cost for everyone (even if two classes get 1-2 more feats than others, in practical terms it matters little). Still I couldn't say if a sidekick is worth less or more than a feat...
Why not mandatory?
I don't even know what is the final version of sidekicks in the upcoming book, but in general I think it would be best not to try and see as something that is part of the character: if you do that, immediately the game is seen unfair if some player get a sidekick for their PC and others do not. So then everyone will want a sidekick, and the DM has double the amount of characters in the game.
I’m not sure that some characters getting Sidekicks and not others is any more unfair than some characters getting spells and not others, or some characters getting to throw a fistful of d6 at any enemy one of their allies is adjacent to and not others. Class-based design inherently gives certain features to certain characters and restricts those features from other characters. That said, giving everyone Sidekicks might still be the way to go if you wanted to do this, and in that case I think it’s safe to assume that DMs who are interested in the option are comfortable with doubling the party size. Otherwise they just wouldn’t use this house rule.
The real point of sidekicks is explicitly to help a party with too few PCs (in the Essentials it seems it's even suggested only for "solo" games), without the complexity of a whole second PC for the player.
Agreed! Of course, the fact that WotC considers them simple enough that they could be played alongside a full PC by a single player who may be playing for the first time was what gave me the idea that they might be simple enough for all players in a party bigger than 1 to control their own Sidekick.
Instead, I have simply allowed experienced players to play 2 PCs at the table, when the number of players is small. It's not a privilege for such player, it's a burden to manage two whole characters, each one takes a single share of treasure and XP. Sidekicks can help lessen the burden, especially if the sidekick is chosen to be a spellcaster in order to fill a gap in the party's capability.
Since in both cases the motivation is to increase the party, there is no need to have the player pay a price by reducing the first PC's features.
Make sidekicks a character option, and competitive players will look at it as an opportunity to be better than other players, if the "price" is right. Kind of a self-fulfilling prophecy. The problem is that it's very hard to set the price right. As I said, I would start with something that at least is shared among all classes: feats, proficiency bonus or even XP. Thinking of sidekicks in terms of action economy is interesting, but problematic once you consider that some sidekicks may be useful only in combat while others may be designed to be maximally useful outside of it.
I’m not suggesting this as something players opt-into. I’m suggesting it as a rules hack, where certain classes (possibly all classes, if that’s what seems like the best option) get Sidekicks as inherent class features at certain levels. Since that would dramatically increase party damage output, I figured putting the Sidekick feature at the levels where a similar jump in damage output is already expected, and replacing the source of expected damage increase by the default rules, would be the easiest way to implement the hack.