Establish PC's goal in the battle
Never forget that the battle has a purpose but your PC's motivation might be different. Are they fighting to help win the battle to save a city? Are they forced to fight and want to survive together? Are they protecting an NPC friend who will fight no matter what? Do they defend their honor after swearing that the right flank will hold? The battle itself cannot be the goal, there has to be a well-defined challenge for the party.
Assure that players have competence to attain that goal
If they are just fighting to win the battle, they need something to differentiate them from a sea of regular troops. They can be commanders or officers, they can have their own cohort to lead or they could just be veterans instilling hope and courage in others. Otherwise they will have no agency.
Conversely if they wish to attain a different goal, such as defending their friend prince from death, they need to be put near him - or have at least a way to reach him.
Treat the battle like a scene that surrounds the players
It's easy to forget that the battle is just another way of shining light on the characters. Your players are assumed to fight near constantly, unless they can manage to disengage the opponents. Do not treat that as combat encounter, but rather as a puzzle that can be solved through leadership, cunning and violence. Instead of having your barbarian use his rage to fuel his damage output, let him burn it as a mean to attain advantage. So, using that as an example if your players protect an NPC - Alice, The Brave Maiden from death in battle, Brong the Barbarian might spend some of his rage to clear an opening to reach her in time.
Make battle events relevant to the players in their own scope
Which means that as the battle is going they seem to hold their own, but when the enemy sends reinforcements, they might have to fight an elite enemy as if it was a standard one-on-one encounter. If they rally their troops to exploit an opening, describe how lowly soldiers ravage enemy lines with their newfound courage. Also, limit their knowledge - they shouldn't have any idea what happens on the other flank, unless party rogue climbs the remains of a catapult to scout the situation.
Ensure party goals can be met regardless of the outcome of the battle
That's pretty straightforward. Give players an opportunity to carry mortally wounded Alice into the forest. If their flank collapses, let them be warned to cast invisibility to escape. If the fortress is being overwhelmed, let them carry the MacGuffin to safety in the nick of time.
Last but not least
Assume they act as a team like they would normally do
That means protecting each other, keeping formation, using their basic fighting skills to attack and defend. Don't force the players to tediously adhere to the battle grid or roll dice all the time. Just assume they are fighting a battle and throw in a juicy description how Cerion the Fighter shield-bashes a particularly large orc and Daria the Mage blasts him into a group of goblins. If you want to portray battlefield attrition, you can abstract their actions to a single "performance" roll, which would penalize them with wounds or depleting battle supplies for a bad roll, e.g. Daria would use one of her spells or Cerion would lose some hitpoints. You may even let them haggle with you to get out of trouble, offering to use up a spell or lose a piece of equipment instead of being wounded etc.
Mass battles usually abstract mobs as if they were single units. You have to do the opposite - abstract all the fighting chaos into landscape events not different from weather or sunset and have particular bite-sized challenges appear from it.
Refer to fictional battles as reference material
And the classic example of that is Battle of Helm's Deep - let's stick to the film version.
The characters have a goal - to protect people of Rohan in the fortress (and to win a kill count contest). They have competence and a role - they are leaders and champions, even though the battle is commanded by Theoden and elvish force by Haldir, they have an influence over troops around them. They fight all the time, but only interesting bits are shown on screen. They only are aware of their immediate surroundings and what happens at the gates is not shown while they are in the keep. They in the end achieve their goal and even if Eomer wouldn't come, they would still have hope for women to escape from the caves. Finally, they meet specific challenges - to hold the wall first, then to kill the famous Torchbearer Orc which fails, then to hold an opening for the elvish troops to regroup. When a battle takes turn for worse, Gimli and Aragorn heroically defend the bridge allowing soldiers manning the gatehouse to rebuild, and afterwards protect the king while he retreats to the keep. Then they convince him to lead one last sortie, while arranging for the women to flee through the mountain pass. Plenty of small-scale challenges in the midst of total chaos and what a battle to behold!
There is no easy answer
This is a tough situation and the solutions are all generally things that can make players feel that their strategies are being specifically targeted. That doesn't leave you with a lot of options for how to address, but here are some considerations in evaluating what to do.
Talk it out
As has been discussed, talk about what's happening with the players. Managing so many creatures isn't fun for you, and that's totally reasonable. Let them know it's a cool tactic and can be used, but please don't use it all the time. Chat about what they like about it, what concerns you, and at least see if they can understand where your coming from and see if they'll choose to alter their strategies.
I had played a bard for awhile that used animate objects and it honestly got tiring for me, too. And it seemed too much, so I only used it when it really made sense to use it.
Summoning is tricky, the DM technically picks
Going by pure RAW, the character's aren't picking the creatures, the DM does. But honestly, that's not a lot of fun. I don't think I've played at a table where the DM has picked the summoned creatures. Using this in your discussion may be a reasonable tactic to show that if you wanted to press the rules-first approach, then you could allow them to summon, but that you choose (maybe randomly) the creatures. That limits the capability within the rules, but it definitely isn't quite the same fun/feeling for the players.
Nerfing the spells
I'm really not a fan of this, especially if the strategy and use is by the book (which it kinda isn't with the above, but you get the point.) Taking away toys because you don't like it can present it's own table issues. There are better ways to handle this.
Encounter design
This ultimately is most likely your biggest lever here. While you don't want to create every encounter that counters this strategy, it isn't crazy to start filtering them and also having the minions of the BBEG know the strategy to counter it.
Counters
The most obvious here is going to be bringing in monsters that are resistant or immune to mundane damage. Summoned creatures aren't usually dealing damage that bypasses magical damage resistance or immunity, so bringing monsters in with those traits nullifies the summoned creatures strategy.
Next up is area affect attacks. These summoned creatures generally also don't have a lot of HP (especially the tiny animate objects). Drop an AOE on the, and you'll wipe them out.
Environment design can also play a part here. Make it so it's difficult to maneuver or have room for the summoned creatures and the option to summon them gets taken off the table.
Keep everything as-is, but introduce Handling Mobs
Chapter 8 of the DMG (Thanks goodguy5!) offers some optional rules around Handling Mobs:
Instead of rolling an attack roll, determine the minimum d20 roll a creature needs in order to hit a target by subtracting its attack bonus from the target’s AC. You’ll need to refer to the result throughout the battle, so it’s best to write it down.
Look up the minimum d20 roll needed on the Mob Attacks table. The table shows you how many creatures that need that die roll or higher must attack a target in order for one of them to hit. If that many creatures attack the target, their combined efforts result in one of them hitting the target....
If the attacking creatures deal different amounts of damage, assume that the creature that deals the most damage is the one that hits. If the creature that hits has multiple attacks with the same attack bonus, assume that it hits once with each of those attacks. If a creature’s attacks have different attack bonuses, resolve each attack separately.
This attack resolution system ignores critical hits in favor of reducing the number of die rolls. As the number of combatants dwindles, switch back to using individual die rolls to avoid situations where one side can’t possibly hit the other.
I haven't personally used this before and, as always, talk to your players about it. This isn't about you as the DM using this for your monsters, but minimizing the player interaction with their summons. This may not be what they're looking for.
But having fun is the key
Balancing letting your players use the tactic they enjoy with challenging them and yourself is the name of the game. Let them shine, but also put them in situations where their go-to strategy isn't an option. Coming up with new strategies and working out how to handle an encounter differently can be fun, too.
Best Answer
In case no one here has first-hand experience using those rules, here is a Reddit post discussing the topic and it's flaws, as well as an alternative.
The OP there argues that the system is very slow, players don't have that much control on the morale of their allies and focus is withdrawn from the players because of the complexity of handling the armies calculations.
You might consider using War Machine, an old system, but one that is reportedly well suited for 5e.
An excerpt of the post: