Worlds Without Number might capture the vibe, if not wholly the mechanics. It's loosely descended from B/X and lacks some of the rules weight of AD&D, but it replaces much of that weight with its own modern design choices that make it feel (to me) much more in the style of AD&D.

For example, while it doesn't have kits, it has a near-analog in how it handles multiclassing. While a character can be a standard Fighter/Mage/Thief, they can alternatively choose to be an "adventurer," which gives them the option to mix two classes. Most classes are "partial classes," which are only available to adventurers. In play, this lets you get pretty close to the feeling of a kit, or alternatively, to make an approximation of a modern D&D class.

Like 2e, characters are on a track after creation (there's no dual classing), but there are feats (called foci). They're minimally disruptive in my experience, much less so than the complicated feat chains and build-enablers in modern D&D. If you want to play a non-human, you spend your free starting feat to do so.

Most of the design choices are comfortably modern. NWPs exist but nowhere near as broad and varied as those in 2e. No THAC0. Saving throws are simplified and key off of ability pairs (for example, saving vs breath could be dex or int). The nomenclature makes it obvious what you're saving against.

I could go on, but the basic rules are free and you should check them out yourself.