They are definitely a PE firm. They buy up struggling companies with the aim to revitalise them, or otherwise recoup the cost of investment+ profit. They have switched to mainly relying on traditional debt rather than outside investor money recently but that doesn't make them not PE.
In fact this is much like the older form of PE, where efficiency gains were the main objective.
Bigger PE firms now usually focus on roll-up strategies (buy loads of similar companies and merge, say car washes is big right now for example, as well as dental, vet and family doctor/GP practices) as well as utilising bucket loads of leverage to amplify gains. This does not however make what bending spoons is doing not PE.
"They buy up struggling companies with the aim to revitalise them, or otherwise recoup the cost of investment+ profit."
1) Nope, they are focused on taking advantage of customer lock-in to raise prices, while reducing operating expenses to increase cash flows. There may be some initial reinvestment to increase surplus of its users, before raising prices substantially. 2) "recoup the cost of investment+ profit"? Yeah lets see if that pans out. The acquisition price is assumed to be under a going-concern basis in perpetuity, if they muck things up with the choices they make the acquisitions have a limited life to increase and capture those cash flows to deliver a positive NPV investment. The demand for the firms products are not perfectly inelastic w.r.t to price.
But PE firms don't have their own workforce, Bending Spoon does, which is why their model differs from, say, Apollo.
The fact they use some of the same tools doesn't mean they are doing the same thing. The majority of Blending Spoon's employees are devs, not finance people.