RAW, I would say yes.
At my table, I would probably be less strict and allow a Beastmaster to train their companion to do more tasks in combat, with some limitations.
The basic principles of 5e rule interpretation should apply here, namely that things do what they say they do and not more. There is a list of things a Beastmaster can command their animal companion to do, and that list is shorter than the list of actions that a generic creature can take in combat.
An animal companion is part of a team with its Beastmaster. The Beastmaster trains the animal companion to follow specific commands and, importantly, to act at the Beastmaster's direction rather than on its own. So it's less that the animal companion cannot do other things and more that its training is limited to a finite set of maneuvers which the companion can reliably understand and execute in a combat situation.
That training isn't represented in-game (you don't really get incrementally "better" at handling your companion), and consequently there are no mechanics for adding new abilities to the repertoire.
So a Beastmaster can direct the companion and tell it to attack or dodge (familiar behaviors, attractive to generic Beastmasters, that could be practiced repeatedly during downtime). But giving it instructions on something more abstract might be problematic, and even though you could make a plausible case for training an animal to do very specific things with objects the lack of a corresponding game mechanic makes this impossible (RAW).
That's not to say that the only thing an animal companion can do is obey commands, and if the normal cues it gets from its Beastmaster aren't coming it'll have to make its own decisions. A baboon, struggling to protect its incapacitated friend, might fumble with an object. Maybe it grabs a fallen sword, having noticed that the enemies were wary of the object when its friend held it. Or maybe it has seen someone use a Wand of [something], and might try to make it work as they observed.
Commands to use these objects might fail because they require abstract understanding the baboon doesn't have (like "grab the sword" would probably mean "grab the sword by the handle and get into a reasonable stance", but that's too much to convey in a <6 second command).
If allowing a Beastmaster to continue personally direction its companion (through the player), expanding the possible actions available to the companion could be problematic and open up some exploits. It becomes a lot more like having a backup NPC available at all times, and some players might purposefully incapacitate their characters in safe locations just to trigger the extra abilities in their companions.
As very basic examples, a downed Beastmaster could simply direct its companion (nominally acting on its own, but still controlled by the Beastmaster's player) to feed them a Potion of Healing (or similar) to immediately get back up. Or, in a room that might be laden with traps, the companion could be used to scout in a way it normally could not and bear all of the risk while the players hang out in the entryway.
It's not necessarily a huge deal, and I think that meaningful exploits would be rare and not necessarily desirable. But it's awkward, and there isn't a particularly strong case that this was an intentional transition (as opposed to an inadvertent edge case).
I think that the operative question is really about who controls the companion if the Beastmaster is incapacitated. Having the DM do so resolves all awkward contradictions about the expanded menu of possible actions, and is probably the smoothest way to implement the rules as written on this issue.