I think “lessons will pile up” is the wrong way to look at it.
There is a set number of lessons you have to go through until you’re done with WaniKani. Every one you do makes the pile a bit smaller.
When you unlock new lessons, the pile of things you have to go through doesn’t actually get any bigger. WaniKani just allows you to take some more items off that pile and put them into the pile of things you are currently learning.
It is entirely possible to get overwhelmed by having too many reviews and those absolutely can reach the point where you can no longer efficiently learn (especially if you take a few days off for whatever reason). You can’t get overwhelmed by having too many lessons, because like I said there’s a fixed number of those and they also won’t magically triple if you take a couple of days off.
So you absolutely should prioritize keeping your reviews per day at a manageable number over keeping your lesson inbox at zero.
(Also, a strategy I’m using is, when I’m reaching a new level, I do the lessons for radicals first, then the kanji, then the vocab - because I need the radicals to unlock all the kanji, i need all the kanji to level up, and I can add the vocabulary during the inevitable lull when all my radicals are in the 2-day gap because Apprentice 4 and Apprentice 5. I’m not 100% sure how effective that strategy is but I feel like it’s a good compromise between a) leveling up as fast as possible and b) not doing everything at once. Of course you shouldn’t start the new level’s lessons before you’ve done everything for the old one.)