First, you seem to be starting from the assumption that all words that are similar between Persian and other IE languages must be cognates. But that's not true; there are three distinct reasons two words can be similar:
- Actual cognates. In your first example, "Tochter" and "دختر" ("duxtar") are both direct descendants of PIE *dʰugh₂tḗr.
- Pure accident. Your second example, "bad" and "بد" ("bad"), is actually the standard example of this, but your "Спасибо" example might be more fruitful. The Russian word is a relatively recent contraction from "съпаси богъ" ("sŭpasi bogŭ", "save us God"), which has nothing to do with the PIE root *speḱ "to see" that the Persian and various other words for gratitude ultimately derive from. For actual western words related to "سپاسگزار", you have to turn to things like German "spähen", which you'd never guess were connected at first glance.
- Borrowing. Armenian "վատ" ("vat") sounds a lot like Persian "بد" ("bad") because it descends from a word that late Classical Armenian borrowed from Old or Middle Persian. (There are of course also borrowings into Persian from other IE languages.) If you tried to reconstruct "վատ" back to PIE following the rules of Armenian sound change, you'd get garbage, even though it actually is an IE word—because the first half of its evolution happened in Persian, not in Armenian.
So, what about your question? I don't know much about modern Farsi, but I think it's more likely that it didn't have a casual word for "yes" until one was invented, not that it had one, lost it, and invented a new one.
Trying to guess a priori what root an unattested word might have come from is a mug's game. You're expecting to find "yes" in a word for "to want"—but the actual roots IE languages' "yes" words (both modern and ancient) come from have meanings like "already", "to see", "itself", "this", "that", and "therefore". Or they're contractions originally meaning things like "so may it be" or "this is". Or they're borrowed from neighboring families and can't be reconstructed natively (try to work out what Romanian "da" could derive from in Vulgar Latin…). None of this is anything that could have been guessed; it all has to be found, or reconstructed by the comparative method. (You might be able to guess a few of the colloquial words that derive from "hail"-type interjections. But then you'd find dozens of false positives for every correct hit.)
Most IE languages' modern words for "yes" were coined or borrowed relatively recently. Other than the Germanic and Slavic languages, most IE languages answered in the affirmative by repeating the verb, as in Latin, or modern Irish. Of course these languages did all have words with some kind of positive/affirmative meaning, but they couldn't be used as casual "yes". And they aren't related to modern casual "yes" words—e.g., Latin's "enim" ("truly, so") is related to all kinds of things, like Baltic "anas" ("that"), but not to any modern Romance word for "yes".
Germanic *ja and Slavic *da are cognate—both come from PIE *yē, which means "already", and has descendants all over IE like Latin "iam" and the Iranian family of *hyáh- relative pronouns. Exactly how *ja and *da are related beyond that isn't clear, but obviously Iranian languages never used those *hyáh- words as a casual "yes".
Finally, your proposal kind of contradicts itself. For example, you propose a lost Iranian word that isn't related to "ja"—and then, to argue for it, you point out that there's a modern dialect word that isn't related to your own proposed word, and that does sound like "ja"? That would be an argument against your theory, not for it.
