When I think of rek'sai, I don't think of her as a champion who would want to be in a long, extended fight. She either does her combo and gets you low enough to kill you, or just manages to chunk you and has to run away after.
depends on whether you're talking about pre or post-rework rek'sai. originally, she liked longer fights better. that's just a matter of preference I suppose. I like her current kit waay better, but I think bruisers as a class should enjoy drawn out fights more.
its also worth noting on whether we're talking about a small update or JUST reworking the passive and THATS IT. my theoretical passive change probably wouldve come with a few other teeny changes (like ezreals update)
your passive idea is a good starting point, but I think its a bit too similar to Phase Rush. If you wanna make a passive that fits in with her current kit, perhaps keep the current healing-for-fury effect but buff the values a bit. then, make it so that if she reaches max fury; maybe she gets the movement speed you said, and the healing she gets once burrowed is buffed significantly. maybe even like... have her burrow cooldown reset to zero when she hits the maximum fury (as a small buff because I hate hate hate that her burrow cooldown is so long) and have the Max Fury healing come faster when she burrows.
basically; that adds up to: hit max fury stacks, get a movespeed boost, get your burrow cooldown reset, and get really big healing when you do burrow. that'd help the in-and-out playstyle you suggested.