The answer to the specific case you suggest is no. The answer to the overall question, sadly, is yes.
Spell addict does not get you Epic Spellcasting
Strictly speaking, at least by Wizards’ errata rules, the spell addict doesn’t actually get the eight spellcasting levels indicated in the table. The text, which takes precedence, says
he adds the levels of spell addict to the levels of some other spellcasting class the character has, then determines spells per day, spells known, and caster level accordingly.
Nowhere in the block is it mentioned that the odd levels of spell addict count double, and it specifically says “the levels of spell addict.” But we can assume that was a failure to review what they’d copied and pasted from the SRD.
Even ignoring that, you only get the spells per day, spells known, and caster level that the chosen class would have gotten. Wizard (or whatever) 21 gets a higher caster level, and two new free spells scribed into the spellbook, but the class doesn’t grant Epic Spellcasting: you have to take the Epic Spellcasting feat for that. Wizard 21 doesn’t even grant a bonus feat, not that spell addict would give that to you anyway (“He does not, however, gain any other benefit a character of that class would have gained”).
Moreover, having Caster Level 21st isn’t even a requirement for Epic Spellcasting, but as Brian notes, 24 ranks in two different skills is. Spell addict does not accelerate skill rank maximums, so at 20th level you’re still limited to 23 ranks: you don’t qualify. There are, however, ways around that.1
That doesn’t matter, because in addition to the prerequisites listed for the feat, Epic Spellcasting is an [Epic] feat, and they have this rule:
At 21st level, and every three levels thereafter, the character may select an epic feat in place of a nonepic feat.
Prior to 21st level, epic feats aren’t even an option.
Enter the dragon
There is, however, a way around that; sadly enough, this has nothing to do with the spell addict or any other poorly-considered third-party material, and is instead poorly-considered first-party material, of a most traditional variety: once again, Wizards of the Coast completely ignored the myriad options that PCs have for becoming monsters.
Draconomicon page 66:
These feats are available to characters of 21st level or higher. Dragons of at least old age also can choose these feats even if they have no class levels.
Huzzah, dragons get to ignore the rules, and shock of shocks, PCs get to be dragons. Draconomicon, presumably, knew that bit (since it discusses it a fair bit), but probably assumed that doing so would mean eating a lot of RHD and LA. Unfortunately, there are plenty of spells for becoming a dragon, and then re-arranging or changing your feats. Since you are a dragon while selecting them, you get to use this rule. And you don’t need any RHD or LA.
Of course, why should you even have to go through that much effort? Races of the Dragon introduces the Dragonwrought feat, which changes a kobold’s type into Dragon, and then gives it the standard draconic age categories – including old. It even explicitly says that dragonwrought kobolds take no penalties for age, so going straight for the +3-to-all-mental-scores is optimal anyway. Turns out you also get to choose Epic feats.2, 3
You do still need to meet the prerequisites for the Epic Feat you want to choose, which for Epic Spellcasting means 24 ranks in two different skills, and 9th-level spells. The already-linked early-entry handbook has ways around those; another character could get you the skills for you by 13th level.
Keep in mind that Epic Spellcasting is utterly game-breaking even if you get it as an Epic character – you really should never take the feat, but I suspect that you will literally never find a game that will allow you to take it early. At that point, you should stop pretending that you’re playing Dungeons & Dragons at all, and just break out your favorite god-game system.
The section marked “2)” that’s mostly in red is the one relating to getting skill ranks above their normal maximum. Almost no tables will allow you to do any of them.
Note that this has nothing to do with whether or not a dragonwrought kobold is a True Dragon – Draconomicon opens up Epic Feats to all dragons, true or not.
Ironically, if we ignore early-entry tricks, this isn’t actually all that powerful. Most Epic Feats require things you cannot have pre-Epic anyway, so those aren’t available, and very-nearly-all of the feats that don’t have such requirements, are really weak to begin with, and never deserved to be Epic in the first place. No game ever is going to be broken by a 6th-level ranger who has Combat Archery, for instance.