Could English kings actually just legitimize bastards, like in Westeros?

Yes, they could! Although it gets really really weird, in the best dynastic scholarship way.

So, let’s talk English inheritance law! In Saxon England, all sons of a King were titled as “aethelings” and were eligible to inherit regardless of their legitimacy. The Papacy was not a huge fan of this, being rather a big proponent of the idea that Christian marriages should be important to monarchs and future monarchs, and tried to outlaw the practice, not always successfully. King Aethelstan (924-939) was a bastard, as was William of Normandy. But gradually succession through legitimate union took hold, sort of…

For a while, you had something of a mixed case, where royal bastards were given the last name of Fitzroy (son of the king) and wore royal coat of arms marked with a bend or bar to distinguish them – as we see above. Especially in the reign of Henry I, there were about 21-25 Fitzroys running around who were very very powerful people with Earldoms and Dukedoms, and while they weren’t guaranteed a place in line, they could sometimes have one. Robert Fitzroy Earl of Gloucester was a potential claimant for the English throne during the Anarchy, although he ultimately ended up backing Empress Maude over King Stephen instead.

You then scoot down to one of the weirder bits of dynastic tomfoolery that took place during the Wars of the Roses, and how it is that the Tudors wound up with a claim on the English throne. John of Gaunt, richest and most hated of the sons of Edward III, had a bunch of children with his mistress Katherine Swynford and then married her. The ex-facto results of this union were declared legitimate repeatedly by Kings, Parliament, and Popes, as quid-pro-quo for supporting Richard II, although the condition of legitimacy was that they had to give up their claims to the succession.

When Henry IV usurped the throne from Richard II, and was feeling insecure on his throne, he recinded the titles that had been given to the Beauforts through their legitimation, as a symbolic underlining of the situation, and got  the succession re-ordered in Parliament – although to keep them sweet, the Beauforts were given the lands of Owen Glendower, which is where the first Welsh connection comes in. A bit later, Henry VI didn’t have much in the way of close relatives, he did something very odd: he legitimated the Welsh House of Tudors, who decended from his mother’s second marriage to Sir Owen Tudor, in 1452, and then in 1455 married Margaret Beaufort to Edmund Tudor, linking the two families

The dynastic chaos of all of this seemed to have left an enduring lesson after the Wars of the Roses. While there would be powerful royal bastards – Henry VIII’s son Henry Fitzroy Duke of Richmond and Somerset, Charles II’s bastard James Fitzroy the Duke of Monmouth – they would never be added to the succession, even if it meant enduring the occasional rebellion (see Monmouth’s Rebellion).