Why tho? I think the Beastmaster is the prime candidate for a beast sidekicks at level 3. I see a lot of people, mainly on Reddit, that act as is sidekicks were just another features they could choose to add to their character. In my book, leveled sidekicks should be treated as rare equipment or magic loot as they make a character more powerful than the game assumes.
IMO, if those rules actually make it into prints, the sidekicks will only level-up every other character level (making them lvl 10 class, if you want). So a Beastmaster (or a chain warlock, why not?) would gain a lvl 1 sidekicks at level 3 and their class features would give give those sidekicks even more benefits: magic attack, invocations etc. Maybe another character could recruit a sidekick at higher level, but the BM's one will always be better. Sure, if the DM gives at-level sidekicks to everyone at any levels, it may unbalance the game, but that is true of many things.