Since the traditional TKD forms were based on shotokan karate, wouldn't they be designed with pretty much the same applications as Japanese Karate, maybe with minor modifications?

I wouldn't say they were "designed" with anything...they are the same forms. The applications likely weren't even thought of, and any modifications made were probably made without thought to the original application. To find their applications, I would look to a good shotokan school (a not good shotokan school would not practice applications either). Or even an Okinawan karate style which has versions of the same kata, though that may be a stretch.

Although a good Shotokan school might train the applications, they were never part of the original curriculum, Funakoshi didn't teach t any.
Therefore any application practiced nowadays is reverse enginering and so are those of TKD, as a result the applications differ just as much as the form itself differ
This is because they are reverese enginered from how the forms look now
Form preceding application is what does that
If application precedes the form, and the application remain the same, then it doesn't matter if forms are changed to suit the founder, the application is still there

Doing a Pinan/Heian Kata with opponents is not what I call application

Example, Pinan Shodan/Heian Nidan, the tripple jodan uke/age uke, because of the deep rooted Shotokan stances the attacker has to step backwards while attacking, which as practical application doesn't make sence
Now if you take the Okinawa Karate with their higher up stances(shorin ryu) their bunkai where the triple Jodan uke could in fact be 3 attacks to pressure points in the neck erea makes much more sense then they would with the deep rooted stances of Shotokan

I agree. If you want the closest to "real" kata, you need to go to one of the Okinawan styles. The TKD patterns are a copy of a copy of a version of the "real" thing. I was trying to be nice, in giving some shotokan schools the benefit of the doubt. I've never trained shotokan style, so I'd rather not make too many generalisations. I'm already generalising about TKD.

All hail the mighty original undefeatable karate kata! TKD is fake Korea is a copy cat. Only Karate has applications. HA I mean why does TKD even exist? No one takes it seriously right? Just kidding, now by traditional TKD I assume you mean ITF correct? This may be true. WTF TKD has its own very diffrent and unique form set including the Taeguek, palgwe, and WTF black belt forms. Each have practiced applications I assure you. TKD KARATE one is not better than the other stop the generalizations please.

CHUCK NORRIS invinted all kata anyways, he tought Funakoshi everything he knew.

I see what your all saying, however shotokan was based on okinawan karate so logically, I would htink the same applications would be burried in there somewhere. And I'm not saying TKD is bad or no good or anything, I like TKD, that's what I train in, I'm just looking for the connections between Okinawan, Japanese, and Korean martial arts traditions.

I have a 2nd dan in karate (shotokan) and I also have a dan grade in aiki jujutsu and at the moment i am training in tae kwon do ITF, currently blue tag. there are many similar movements/techniques in the tuls of tkd as in shotokan kata as shotokan influenced TKD.

Since doing aiki jujutsu I have been able to understand and study my karate kata and there are thousands of jujutsu techniques that relate to kata movements/techniques - and this is also true of TKD patterns. it's just that in some modern dojo's/dojangs they are not taught, probably because they are not known - e.g - this is a block, this is where you kick or punch.

I am at present working with my TKD teacher to introduce a self defence syllabus within the club and they are mainly all from jujutsu techniques which are found within kata/tuls - it's all there, just have to look a bit deeper and analyse it.

My karate teacher told me once - I teach you karate - you then go and work on it yourself, your own applications - they will be yours. of course some will be good and some will be bad - that's where you test them.

as the self defence federation would say - CREATE, ADAPT, APPLY -

Well, I'll keep that in mind when working on forms and stuff. I've slowly but steadily been finding more and more applications, very effective ones at that. So it's a work in progress.