This is an example of one of the downsides of D&D as a whole... because it evolved from wargaming, there have always been precise measurements and graphical interpretations given to all combat, skills, and abilities. Thus players have been trying to manipulate all the physics and math of the ruleset to their advantage over the years, and the designers have always tried to stay a step ahead. Unfortunately, that has resulted in a more "narrative" way of looking at the game usually being ignored or at least pushed towards the rear.
If you look at the spell purely narratively (like if you were seeing this spell as written in a novel with no game aspects attached), you basically have this invisible floating disc that holds hundreds of pounds of stuff for you, and you can have it follow you along as you walk away. And only the most logical reasons why it can't follow you-- like a door closes between you and the disc, or you teleport away and the disc can't... ever really come up as issues.
So there's something to be said in actually playing the game of D&D in that way... rather than trying to work out and work around game rule mechanics to justify or not justify the actions of you and the disc. After all... is there anything lost if you the PC can push the disc away from you in front of you so that the disc is leading (rather than it following along behind you) because that's what you need to happen narratively in the story? Unless you are a hard game simulationist, I don't think there is. So in my personal opinion... don't worry about it. You have a magical spell that creates a disc that can carry stuff, and you can move it around to accomplish what you need the disc to do.
EDIT TO ADD: Here's the other thing that you gain if you treat a lot of D&D's magic narratively... you can become more thematic in the same way you can in say a game like Champions. In this case... you are a Telekinetic force mage. All Tenser's Floating Disc is, is another way of using telekinesis. Yeah, the game assumes they are distinct and individual spells that do not overlap... but if you look at the ideas narratively... you have a telekinetic force wizard that can easily move small objects with no real effort (Mage Hand); can move larger objects slowly (Floating Disc); can bludgeon other creatures using telekinetic force (Eldritch Blast and/or Magic Missile); can throw up telekinetic barriers in front of them (Mage Armor, Shield); can telekinetically bend light around themselves (Invisibility); can lift themselves up into the air (Jump, Levitate, Fly); can move themselves at superhuman speeds (Phantom Steed) etc.
In Champions, if you select all these different abilities and uses and narratively treat them as from one thematic identity, you get to buy them all at a discount. Unfortunately, D&D doesn't have that, and thus it is way too easy for a spellcaster character to just take all different types of spells randomly without having a theme. Now sure, some players will build pyromancers, or necromancers, or cryomancers or whatnot... but invariably they'll also have a lot of individual non-thematic non-narratively consistent spells too, because the game doesn't prohibit it or make it worthwhile to not select them. So even a devoted pyromancer will have the ability to create a Phantasmal Force illusion, because.. why not? Which makes sense from the gameplay perspective of D&D, but doesn't so much when you think about a wizard narratively. And I sometimes think the game (or more to the point, MY game) might be occasionally better served by treating magic more thematically, rather than "every caster can take every spell" that the game allows.
