No, it can't lift a party across a river
It can create a single rope "bridge" that the party would then need to climb/walk on to cross the river.
Holding vs Lifting
The Rope of Climbing (DMG, 197) does not Lift anything per the description. What it does is:
If you hold one end of the rope...the rope animates. As a Bonus Action, you can Command the other end to move toward a destination you choose.
This is clear in that you are holding one end of the rope while the other moves towards the destination of your choice. Once the end arrives at it's final destination, you (and others) would then still have to climb the rope. It does allow for easier climbing to limit risk by:
If you tell the rope to knot, large knots appear at 1-foot intervals along the rope. While knotted, the rope shortens to a 50-foot length and grants advantage on checks made to climb it.
It is not meant to carry you up. It simply removes the requirements of using a grappling hook and still requires someone to climb it. If it was meant to carry you up, there would be no need for the climbing mechanics or the knotting to assist in climbing.
As for horizontal/arc movement, that's perfectly acceptable as well. There are no stated limitations on how the rope moves 10' and in fact does state:
move toward a destination you choose
This allows the rope to move in any direction you choose.
No language on what happens if you overload
Frustratingly, there is no language on the results of putting more weight on than the rope can handle. Other magic items do provide such effects, but it is not described in the DMG and is therefore under the purview of the DM. But given that it doesn't say the Rope is destroyed, it's more likely that it just can't continue to lift/move if overloaded.
Get creative
There is language in the description that just begs for creative thinking (emphasis mine):
You can also tell the rope to fasten itself securely to an object or to unfasten itself...
You can not only use it like a grappling hook that'll always find it's mark, but it can be a 60' long extension of your hands as well. Need those keys? Rope gets 'em. Gotta pull that lever across the chasm? Send the rope! It is not just a means of getting from Point A to Point B, but also a means to bring/interact with things at Point B.
The intent is probably that only whoever activated it can use it.
functions like a mundane broom until you stand astride it and speak its command word.
Activation requires physical contact. From my reading, this intent is given by the use of you, in every sentence. Usually it would be unclear who exactly is "you", but the earlier sentence defines it (as the person who stood astride and spoke the command word).
You can send the broom to travel alone to a destination within 1 mile of you if you speak the command word, name the location, and are familiar with that place. The broom comes back to you when you speak another command word, provided that the broom is still within 1 mile of you.
The wording, if it was intended that any creature could use it, would probably be "A creature that says the command word", for example.
As an additional conjecture, it would be silly that anyone hearing the command word could mess up the item easily like that.
Best Answer
Speaking a command word does not take an action unless it says so
The driftglobe's description reads as follows
According to the DMG, one of the things that you can do without using an actions is:
Since driftglobe doesn't specify that speaking the command word takes an action or bonus action, it would seem to fall under this category, and thus be treated as a free action