The sphere is weightless only when empty

The sphere and the creature are described separately

A Sphere of shimmering force encloses a creature or object of Large size or smaller within range.

Notice how there are two things described here: the sphere and the enclosed creature.

Here is another example, even more compelling

The Sphere is immune to all damage, and a creature or object inside can't be damaged by attacks or effects originating from outside

So the sphere is immune to damage, but the creature explicitly is not. Clearly, when the rules are meant to apply to the creature they say "creature" and when they say "sphere" they apply only to the sphere itself. If it were otherwise and "sphere" did mean "sphere and creature" then this sentence in particular would make no sense at all.

The sphere is weightless but the creature is not

The Sphere is weightless and just large enough to contain the creature or object inside.

So when it says the above (and again makes the distinction between sphere and creature), it says that the sphere itself is weightless and that it contains the creature. It says nothing about the weight of the creature, thus the spell had no effect on it.

Thus, with the creature inside it, the sphere should have the same weight as the creature itself (including equipment etc.).

The "half-speed" part of the description applies only to the enclosed creature

An enclosed creature can use its action...and thus roll the sphere at up to half the creature's speed.

When it says:

Similarly, the globe can be picked up and moved by other creatures.

I just read that as saying that the sphere can be moved by others, but that it does not specify rules for doing so. To read it the other way would be confusing seeing as the previous sentence does not say anything, for example, about how movement would work with picking up the sphere.

So, creatures external to the sphere will need to use the normal rules for moving another object.

Yes, those're the rules.

That's really all there is to it. Except here's a bit more.

PS: I know D&D isn't meant to be a realistic sumulation, but still.

There are times in the history of D&D when it's tried to be a more-realistic simulation. For your purpose I'd recommend taking a look at racial bonuses/decrements to stats; perhaps the PHB1e?

Many people still play that edition, including this author. It can be fun. It's a different game.

5e, however, made the explicit choice to play a different game which prioritized "gameplay" concerns higher than "simulationist" concerns. If you choose 5e, that's part of the choice you're making.