House Celtigar is one of the three Valyrian houses that moved to the Western most outpost of the Valyrian Empire (The other two being Velaryons and Targaryens). Out of the three of them only Targaryens had dragons so they naturally became the leaders.
We know that only people with blood of Old Valyria tame the dragons, they don't necessarily have to be Targaryens.
Yet, during Dance of the Dragons, when Lord Bartimos Celtigar advised Queen Rhaenyra to attack King's Landing with both claimed and unclaimed Dragons, her reply was:
“It will never come to that,” Celtigar insisted. “The usurper will have no choice but to oppose us with his own dragons. Our nine must surely overwhelm his four.”
“At what cost?” Princess Rhaenyra wondered. “My sons would be riding three of those dragons, I remind you. And it would not be nine against four. I will not be strong enough to fly for some time yet. And who is to ride Silverwing, Vermithor, and Seasmoke? You, my lord? I hardly think so. It will be five against four, and one of their four will be Vhagar. That is no advantage.”
The Princess and the Queen
Why does Rhaenyra thinks it is virtually impossible for Lord Celtigar to claim a dragon? He had just as much Valyrian blood as Rhaenyra did. Although her point seems to be vindicated since when Prince Jacaerys called for all Dragonseeds to come and try claiming dragons, no Celtigar seems to be present. But on the other hand, there were two bastard Velaryons who tried and one of them succeeded (Both were brothers). Not to mention, many other dragonseeds, even the ones who were plainly born with Valyrian blood like Silver Dennis failed as well whereas some who by no means looked Valyrian like the girl Nettles succeeded. Since people of such diverse backgrounds succeeded, with only thing common between them being Valyrian blood, in claiming dragons, why does Rhaenyra believes it is unthinkable that Celtigars might have claimed one as well?