A shield takes 1 action to equip (PHB 146). I'm an new DM and would like to allow that in the first round of battle the PC may use the "interact with objects" time (PHB 190) to equip its shield, if not already equipped.

Reason I want to allow this is that I don't like the PC to always have its shield up, just in case there's going to be a battle.

Perhaps I'm overthinking this, but could there be any reason I might consider to not have this house rule?

It devalues "social combat" tactical decisions.

The decision to wield a shield or not has in-game consequences, from both combat and social interaction perspectives.

Mechanically, shields offer a +2 AC bonus (a significant boost in D&D 5E's bounded accuracy system) and require an action to don/doff (a premium resource in the action economy).

A wielded shield indicates that the bearer is expecting a fight. This probably wouldn't raise any eyebrows if you bumped into a merry band of adventurers running around the wilderness or dungeon, but it has a different social context in more civilised locations. Shopkeepers, tavern owners, and city guards may be rightly concerned to see a group of armed mercenaries roll up like they're spoiling for a fight.

The decision to remove shields is a tactical one. Does the party want to try to pass through peacefully, but potentially leave themselves more defensibly vulnerable if a fight arose? Or are they willing to potentially create conflict in order to maintain their defensive capabilities?

Being able to efficiently don a shield with an object interaction undermines this social combat. With this Homebrew rule, you give the PCs the ability to circumvent these social combat scenarios and still maintain combat readiness. This rule removes that decision making and could produce less interesting scenarios as a result.

Personally, I tend to lean towards the PC/NPC interactions, and that means dealing with the social cost of carrying your shield up and ready for combat while wandering around the tavern, or if they have a shield on them at all while attending a high society event. In this type of story, the dangers are not always HP loss. And "the right action" isn't always a fight. But the players need to decide going into events what kind of interaction they want to have. (Fight or Social) And what kind of trade offs they want to make.

The flip side is the more cinematic world where your dressed in a full tux, a fight breaks out, and you tear off your tux revealing full plate beneath and the PC promptly draws their Buster Sword from whoknowswhere.

Neither story type is "Best". It all depends on what kind of play style you and your players enjoy.

No, You would need to house rule it.

• Specific Rules Beat General

Unfortunately with there being a rule that specifically addresses shields and how they are equipped, they override the "use an object" set of rules. You would be required to find another rule that specifically states something to the effect "You may don this shield as a Free or Bonus action." This would be a RAW interpretation.

From a functional standpoint - most tables ignore micromanaging shield equips.

Functionally, in most games, it is assumed that while adventuring, characters are in full armor, ready with their equipment at any moment's notice. The idea here is that the hero in question is carrying their shield equipped as they traverse the wilderness or dungeon delve. Only if a character specifically states they have slung the shield on their back to keep their hands free would they need to "equip" their shield during any particular moment in combat.

Realistically you will only have to observe this rule for specific thematic scenarios - such as being ambushed in the middle of the night while the fighter in question is asleep. The need to scramble and grab their shield and sword now becomes choice the player needs to make, and can heighten the sense of danger and whether or not taking that extra moment to get their AC up is worth the loss in positioning or attacks.

Another scenario is in the event the fighter needs to use two hands to do something - you remind them they have to drop their shield in order to accomplish it. Then it's acceptable to micromanage and force action economy to re-equip the shield. This again creates a meaningful choice and combat mechanic to the event or scenario.

In conclusion - Only if you are truly rules lawyering / nitpicking the party state at every possible moment would you need to micromanage shield equips (or if the player is carrying multiple magical shields they like to swap out). Adding a single piece of gear a player has to constantly micromanage will slow the game down and potentially cause a focus shift away from the adventure, and closer to a table top combat simulator.