Both framings describe real things, and they're about different risks.
Income risk favors the triplex. Say three units at $1,200 each, $3,600 a month. Lose one tenant and you're at $2,400, so you can probably still cover the payment while you re-rent. A single family house at $1,200 goes to zero the day the tenant leaves, and you cover the whole payment out of pocket. That's the vacancy argument and it holds up.
Asset risk favors the three houses. Your triplex has one roof, often one heating plant, one water service, one parking lot, one street, one school district, one flood map. A bad roof hits all three rents. A new employer leaving town hits all three rents. Three houses in three neighborhoods don't share those. You also get three separate exits with three houses, since you can sell one and keep two, while a triplex sells whole.
The cost side leans the other way again. One purchase means one set of closing costs, one insurance policy, one tax bill, and one roof to maintain instead of three. That's a large part of why small multifamily tends to cash flow better per dollar invested.
The thing that decides it for most people is the buyer pool at sale. Single family houses sell to anyone. A triplex sells to investors and to owner-occupants who want to house-hack, which is a thinner market and takes longer, especially if rents are below market when you list.