I used to be the lead developer on an IVR/PBX SaaS solution. It hasn't been necessary to have dedicated hardware for years (ever since out-of-band DTMF has been a thing). The only reason that companies buy it is because they're getting conned into it.
We had a server through which the conversation (SIP, converted from landlines) went, and enabled the agent to interact with the call through either a web page, or hitting keys on their phone. Transfers, blind or not, holds, recordings, assistance, we could handle all of it. To dumb phones. Literally things that weren't able to display the time of day.
Mobile is doing diddly squat, frankly. Same as softphones. The voice quality over wi-fi is terrible, and what's the point of using a mobile device if it needs to be plugged in anyway. Softphones never took off because call centres couldn't reuse their headphone equipment, and having a headphone's "pick up" button work on a softphone was impossible, at scale. It also meant that actually interacting with the call was tricky (change focus of the window to find the numpad, then either click on digits, or type on the keyboard).
One major game changer, however, seems to be WebRTC. A lot of companies are investing big on it (see Dixa and COLT for example).
I really think this is a generational problem. Callcentres of old want the highly integrated hardware. Train your people once, and then they can become robots. New, modern places want the ability to switch providers. They want their agents to be close to the customer, and even closer to the tracking/reporting software stack, without going through 10 layers of proprietary integration.
WebRTC also means that you can simply talk to the person on your website, while you see what they see, and can walk them through whatever they need.