RAW, I would say yes.
At my table, I would probably be less strict and allow a Beastmaster to train his partner to perform more tasks in combat, with some limitations.
The basic principles of the interpretation of rule 5e should apply here, namely, that things do what they say they do and not more. There is a list of things that a Beastmaster can order his animal companion to do, and that list is shorter than the list of actions that a generic creature can perform in combat.
A pet is part of a team With your beastmaster. The beastmaster trains the animal companion to follow specific orders and, more importantly, to act in the direction of the beastmaster instead of doing it alone. Then it is less than the companion animal I can not Doing other things and more than your training is limited to a finite set of maneuvers that the partner can understand and execute reliably in a combat situation.
This training is not represented in the game (in reality it is not progressively improved in the management of your partner) and, consequently, there are no mechanisms to add new skills to the repertoire.
Then, a Beastmaster can direct the partner and tell him to attack or dodge (familiar behaviors, attractive to generic Beastmasters, which could be practiced repeatedly during downtime). But giving instructions about something more abstract can be problematic, and although 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 it impossible (RAW).
That does not mean that the only What an animal companion can do is obey orders, and if the normal signals he receives from his Beastmaster do not come, he will have to make his own decisions. A baboon, struggling to protect his disabled friend, could search for an object. Perhaps he grabs a fallen sword, having noticed that the enemies distrusted the object when his friend held it. Or maybe you've seen someone use a Wand of (something), and I could try to make it work as they observed.
Commands to use these objects may fail because they require an abstract understanding that the baboon does not have (such as "grabbing the sword" would probably mean "grabbing the sword by the handle and adopting a reasonable posture", but that is too much to convey a command < 6 seconds)
If allowing a Beastmaster to continue directing his partner personally (through the player), expanding the possible actions available to the partner could be problematic and open up some exploits. It is much more like having a backup NPC available at all times, and some players can deliberately incapacitate their characters in safe locations just to activate the additional skills of their peers.
As very basic examples, a downed Beastmaster could simply direct his partner (nominally acting alone, but still controlled by the Beastmaster player) to feed him. Healing Potion (or similar) to go back up immediately. Or, in a room that could be loaded with traps, the partner could be used to explore in a way that he normally could not and bear all the risk while the players hang out at the entrance.
It is not necessarily a big problem, and I think that significant feats would be rare and not necessarily desirable. But it is uncomfortable, and there is no particularly strong case that this has been an intentional transition (as opposed to an inadvertent marginal case).
I think the operational question is really who controls the partner if the Beastmaster is incapacitated. Having the DM do it solves all the awkward contradictions about the expanded menu of possible actions, and is probably the easiest way to implement the rules as written in this topic.