In general, I expected startups to be more immune in the current market due to their leanness and being “dead weight”-free in theory
One of the premises for startups having an advantage over big firms is their ability to pivot. But they still need money to ride out lean times and get through the pivot. A startup making X, when X revenue drops, has to pivot fast or cut spending or fold. Layoffs are one way to buy time (through reduced spending) to allow the pivot to succeed. A large firm making A through Z, when half of them start becoming unprofitable can either carry them through (from profits in other divisions) or make cuts. They don't have to fold, and if they make cuts it will be a smaller percentage of their total workforce because it's from a subset of all their offerings, not their entire product portfolio.
Maybe that large firm can't pivot, but if it's got a few billion in revenue still and is still profitable (or has a large bank account or credit line) then it can ride through much better than a leaner company (potentially).
You also have to determine when a company is lean or "lean". Lean as in healthy? Or lean as in they cut off their own arm and spend 5- or 6-figures a month on infrastructure hosted by Amazon and have no in-house IT expertise, but sure, they know their customers.