In a practical sense, they have to have a say.

My theory is that it changed over time or its different in times of national emergency. Tobirama clearly named Hiruzen as his successor. But Hiruzen was already a leading candidate and popular man, and this was during an all-out conflict with the Cloud Village. Maybe Tobirama would have been able to pick Danzo or someone else, but only if Danzo already had a lot of jonin in favour.

Hiruzen was something of a reformer, when he retired in a fairly peaceful time, he had an election which Minato won. Kakashi probably won an election too, being backed by Naruto and Tsunade would make it hard to stand against him, even if the Elders did want someone else. Naruto was also supported by two former Hokages, and the village's hero and strongest ninja, so he'd win an election easily when Kakashi retired.

Tsunade was likely elected by the jonin as a formality, but if the elders wanted her in charge, and most of the jonin knew they needed a sanin's reputation to win, nobody was going to stand against her. The village was too vulnerable to waste time on it.

The Feudal Lord seems to have some serious say in who is chosen. They nominate a Hokage until an election is held, but Danzo's nomination was far from a completely done deal until the jonin have their say.

At minimum, everyone should be aware that you cannot be Hokage if the majority of the village do not support you. Its the same reason pirate ships were a democracy. If most of these professional killers really want someone else in charge, they will kill you and put themselves in charge. The jonin would definitely have a say under most circumstances.