What destroys Balance is....

When thinks can't be replicated by both teams.

As far as the map itself, the dragon change made the prospect of killing the dragon change entirely. Before, it was simply a large sum of gold. Gold can be earned through other means, so giving up a dragon didn't mean the enemy had a buff that you have no way at all of receiving afterwards (unless the next dragon is the same, but the fact they have the first means they're better suited to taking it than you.)

For champions,when a champion isn't newly reworked or released, there's usually a stat imbalance that makes that champion a prime candidate over others in their class. When Cinderhulk was released, it was so strong on Sejuani that even though Amumu and Malphite have basically the same role and playstyle, she had an advantage over picking them.

For new releases or reworks, they tend to add in mechanics that no other champion even has, and it has nothing to do with the damage, shield, or CC stat on the ability. No one has 10 .2s cooldown dashes or projectile-negation like Yasuo's. Not even Braum's can match up, since he only blocks from the front. No one can come from near-death to heal half their HP and deal 700 damage in an area, while being shielded and stunning the enemies, like Ekko can. Other healing abilities don't damage or displace the caster, and other stuns don't shield.

Granted, allowing mirror matches would solve the balance part of this issue, many players think mirrors aren't fun because it's often the case that if one player gets the lead, they keep it. (Although items, allies, and playstyle can definitely argue against that, seeing as it's the same case even if you aren't in a mirror lane) And if they continue to make changes or new releases centered around being Unique, they'll continue to make changes that negatively impact balance, since Unique means the enemy team can't possibly have a similar aspect on their team who isn't the same champion.

But how they "balance" the game now is through counter-picks. If a champion is weaker against some others, then they consider that balance. Even if the losses are in huge part due to a small handful of hard counters.