While it is true that Sling ceased manufacturing Sling boxes, the remaining stock is still being sold and the later models of Slingbox are still operating and expected to continue operating. At the very least, the later Slingbox models of M1, M2, Slingbox 500, Slingbox 350 have all moved to the SlingPlayer for desktop a few years ago, so it no longer uses Flash/browser. I remember having been kicked off the browser on to the SlingPlayer a few years ago. It was a bit of a pain at first to get the thing installed but eventually it works very very well. I have been using the Slingplayer for desktop the last few years on many many occasions and I prefer it not dependent on the browser it's a better user experience. If you have a later Slingbox model today that you're able to playback on your desktop, then you are definitely must be using the SlingPlayer for desktop because Sling ripped out all flash/browser support a few years ago and required you to download and install the Slingplayer for desktop. In other words, at least those later models of Slingbox listed above will continue to work indefinitely using the SlingPlayer for desktop that is no longer dependent upon flash/browser.
Of course the SlingPlayer app for mobile was a different animal-- it did not depend upon the browser--and will also continue to work with the above-listed Slingboxes. Flash is a horrid security mess and needs to go. We all need to be thankful it will die very soon.
As for earlier models of Slingbox, I can't say with any personal experience, but in many forums it seems those older boxes are being left behind by Sling and were dependent upon the browser for playback. I seem to remember many posts about people complaining that the older Slingboxes they have aren't compatible with the SlingPlayer for desktop introduced a few years ago.