Arsthbt

One of the showrunners in the Behind the Episode interview said:

Only Targaryens can ride dragons

Obviously in the books we learn about the Dragonseed and House Velaryon dragonriders. Have they been removed from the show's history?

No, House Velaryon exists in the show. Corlys Velaryon is mentioned in one of the Histories & Lore videos for Season 5,.

When ravens carried the news of Aegon's coronation to Dragonstone, Princess Rhaenyra summoned her own Black Council. With her was her uncle and husband, Daemon Targaryen, considered by many to be the most dangerous man in Westeros. Few lords supported her, but the greatest of these was Corlys Velaryon, who controlled the largest fleet in the realm, and his Targaryen wife Rhaenys, the old king's sister.

_{Game of Thrones, Season 5 Histories & Lore, "The Dance of Dragons"}

It's also worth noting, that whilst I haven't seen the behind the scenes you're referring to, the context is almost certainly set for those families in Westeros. Of those in Westeros he is saying that only Tararyens can ride dragons. This makes more sense when we consider the full quote:

No ones ever ridden a dragon except for Dany, Only Targaryens can ride dragons.

Now we know this isn't true because we've heard the stories of the past and the other dragons in the show. His quote is in context to present day Westeros and those characters we know of in the show.

You mention two groups of people who have been shown to ride dragons in the book series: House Velaryon and Dragonseed. To see how those two groups fit into the show history, I'm going to address their ancestry first.

Valyrians

Valyrians were the dragon masters of Old Valyria, which was destroyed by a cataclysm prior to the Targaryen invasion of Westeros. From what we know, Targaryen's are the last remaining descendants of these people since they left for Dragonstone prior to the Doom of Valyria.1

House Velaryon

House Velaryon is house of Valyrian descent who has intermarried with the Targaryens countless times, and so they contain the blood of Old Valyria's dragonlords through their Targaryen ancestors. Due to their Targaryen ancestry, the showrunners can be accurate in saying that only Targaryen's (with the implied "and their descendants") can ride dragons.

Dragonseed

What you refer to as Dragonseed are the illegitimate children of the Targaryen rulers. While not Targaryen in name, they are partially Targaryen in blood.

Are Targaryens the only people who can ride dragons? Technically no, the descendants of any of the dragonlord noble houses of Old Valyria could ride dragons as well. Also, plenty of royal bastards have been shown to be able to ride dragons. Due to this, it appears that anyone who descended from the Noble Houses of Valyria (and hasn't had the blood line diluted too much) should be able to ride a dragon. This would include House Velaryon and the Dragonseed. In actuality, the show has not introduced the Blackfyre line (or any other bastard Targaryen descendants) and we do not know of any other Valyrian Dragonlord houses who survived the Doom.

Since all of the Valyrian Dragonlord descendants that we are aware of are of Targaryen lineage, the comment that only Targaryens can ride dragons can still be viewed as true; even in a world where members of House Velaryon and the royal bastards have also ridden dragons.