When looking at the length of combat, there are two sides. How long to run the DM's turns, and how long to run the PCs turns.
The PCs turn time can be worked out as the damage per second of Player time (per SECOND not per TURN) divided into the total HP of enemy monsters.
For the DM's side, you ideally want it to be tight and small. This is why monsters are (ideally) very easy to run.
One trick on the DM's side is to realize you are in control of the monster power budget. So you can and should play monsters sub-optimally in order to make the game go faster. Playing sub-optimal tactics on a monster with more power is just as dangerous as less optimal play with a weaker monster!
So arranging for ideal positions for aoes, focusing fire, etc might make your big monster more effective, you should save table time by making the monster more effective and less tactically tricky to play on your side.
Setting up tactical problems for your players, on the other hand, is great.
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The next problem is having enough rounds for interesting stuff to happen, and the fight not being a long sequence of PCs working out damage mechanics with the situation not changing.
Legendary Actions on solos exist for this reason. The idea is that the monster changes the tactical situation multiple times per turn, which should help keep the players engaged and interested.
It also spreads out the monster's damage budget over (game) time, allowing response from players, and not "monster goes, one PC dead, your turn" if you focused it all at once. A monster that dropped a PC per round, but died on round 3, would be a balanced fight in a sense (PCs reliably win), but might not be what you are aiming for in a fight.
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So we want fast to play monsters that pose tactical problems for PCs, not DMs. We also want a dynamic battlefield, one that doesn't get locked down and ends.
My suggestions:
1. Tweak the dragon so it isn't doing the AOE breath alpha strike. Instead, on each legendary action, roll 1d6 and see if it inhales. On a 6, it prepares to use its breath weapon.
2. Ensure your boss has mobility baked into its legendary actions. "The dragon moves its speed and makes a claw attack on up to two different creatures. The claw attacks can be at any point during the move, and a creature attacked by the claw during this legendary action cannot make an opportunity attack on the dragon."
3. Have legendary resists. For extra fun, tie some mechanics into using them to keep the battle dynamic.
Legendary Resists: The dragon can choose to succeed at a save instead of failing one. When it does so, it must suffer one of the following consequences. It can only choose a consequence once before completing a short rest.
* Wing shield. The dragon's wings are damaged, and it cannot fly nor wing buffet until it completes a short rest. If it is flying, it falls and takes 1d6 damage for every 20' it falls, but so do all of the creatures it falls on.
* Bloody eye. The dragon's vision is impaired. It can no longer make perception checks as a legendary action, creatures more than 30' away from the dragon at the start of its turn have advantage on saves the dragon imposes.
* Wounded claw. The dragon can now only make 1 claw attack instead of two.
4. Bake in recovery to legendary actions. "When the takes a legendary action, if there is an effect that would end by the end of a creature's next turn, the effect ends after the legendary action. If it could end after a saving throw, the dragon may make a saving throw after the legendary action. A failure on that saving throw has no consequences."
"Legendary Action: Recover. The dragon may take this action even if unconscious or otherwise incapacitated. The dragon gains advantage on all saving throws until the end of the next opponent's turn."
5. Give it more HP, and more spread out damage. HP is a measure of how long the players spend dealing damage to it. As you want 3+ rounds, you need enough HP to soak 3 full rounds of players pounding on it.
6. Mix AOE damage with spread out damage and at least one "spike" damage. Make applying that spike damage optimal difficult, and be the player's goal.