Uragan is a big problem, because it's chassis is a huge pain to work with. Ukraine even went as far as rebasing the firing mount onto a new armoured vehicle. In Russia, the plan before the SVO was this (even though the "bigger" platform vehicle is weird). Uragan-1M, as far as we know, is dead.
They won't be making promotional videos of these systems if they were duds. The MoD has made at least two videos for them already, and I refer to such videos more as 'honoring' videos. Uragan appears to contribute significantly to the war effort even with it's normal ammo.
Based on the videos, the Uragan does deploy and undeploy, then move fast enough. The long platform might be a problem for overland travel, but they did use a similar platform for the Smerch and Tornado-S. As for Ukrainian variant, attempts to indigenize the platform usually begins with the transport, given that Ukraine lacks the IP to manufacture the original. We see similar things with all the different Grad cousins manufactured in Eastern Europe, each having their own transporter. For sale sakes, each will claim they're better than the original.