You are obtusely refusing the fact that specializing in a particular element is a common fantasy archetype that many D&D players are likely to want to replicate for their PCs. The Elemental Adept feat is clearly intended to facilitate elemental-themed spellcasters staying on theme, but your solution continues to be, "Just break from your theme." Should Ice King cast fire spells? Should a Waterbender have fire-powers? Does Melissandra the Red Priestess keep a cone of cold spell in her back pocket for when fire magic just ain't working? How many more examples of elemental specialists would you like before you agree that this is a common character theme?
Elemental Adept might not work as intended, but your solution--"Don't play an elemental-themed character,"--is entirely unhelpful. Arguing that you shouldn't need to houserule it is the real non-sequitur here.
Why does one need to house rule it at all? I think that it works exactly as intended.
If a player wants to play Ice King and wants to take that feat, have at it.
I have never met a player who wants to play Ice King and absolutely refuses to take Magic Missile because it is force damage or Wall of Stone or Chain Lightning or other D&D spell icons, but if you want to restrict yourself that way, have at it.
The feat does what it is designed to do: bypass a specific elemental resist.
I think that this is a very corner case ability, but that does not mean that one has to house rule it.
You seem to be the one that insists that it has to be houseruled. I have never once said that. I said that a theme based PC would be better off having a backup different elemental spell. If the player does not want to do that and does not want the feat because it might not be used frequently, THAT'S OK. The player can focus his spells and feats/ability score increases however he wants.
As I said before, taking this feat is not necessary, even for a theme based PC. Houseruling it is definitely not necessary.
You seem to be the one stating that not only is this feat necessary for such a PC, but that it is too weak and must be house ruled as well. Talk about entitlement.
If a player is really into playing the Ice King and he really really wants to make sure that his ice damage always works at a decent level, then he will love this feat. No need to houserule it. Mathematically, it is typically not a good choice, but it can be a good choice if the player is happy with it.
Personally, I wouldn't play such a restrictive PC, but I do understand that there are people who would. I personally think that a feat that would allow him to cast any elemental spell as an ice spell would be a better feat for such a PC. Elemental Adept reeks of a mechanical solution (bypassing resist) shoe horned into a feat, but if a player wants to take it, go for it.