Did the King really promise...?

Yes. Alicent is a moron. She basically even said to Viserys "wtf are you talking about?" and he never answered her. She's making shit up.

Note that this plot point about Alicent wishfully misunderstanding Viserys's death-bed ramblings - which is a litle goofy let's face it (even Aegon thinks it's likely crap!) - is *not* in GRRM's Fire and Blood. In addition, in F&B Alicent at the Green Council is very explicit that her reason for denying Rhaenyra is the illegitimacy of R's children. That is, in F&B Alicent reads as much more calculating and hard-nosed and less torn than she does on the show. In general the show often heightens plot points from the book - so, e.g., Ser Criston kills Laenor's gay lover in the middle of a hugely important wedding not during the jousting tournament associated with the wedding. What happens on the show is often much more exciting (e.g. 2. Rhaenys and her dragon smashing through the floor, killing hundreds at least in the last ep.!) often at the expense of plausibility/credibility (Criston should have experienced some consequences both for murdering and for interrupting the wedding; Rhaenys and her dragon should have been crushed by thousands of tons of stone falling on them; and so on).