Well, the Soul Eater and the Rune of Punishment are considered cursed in their respective games and god knows Ted wants to get rid of it in IV despite the fact staying on the Fog Ship will still give him the immortality part, so I don't think it's a simple matter of just "Immortality sucks". Ted vehemently tries to not get attached to anyone and as we see in Suikoden I, the Soul Eater's power that the bearer can use, are directly linked to souls that carry an emotional attachment to the bearer, so I don't feel it's far fetched to believe the rune purposely targets those closest to the rune bearer.
Though it's apparent that mastery over the rune may negate this problem as the TC theorized. Its not like Ted in Suikoden I has problems being around Tir and his family, likewise, Ted's Grandfather has no issues running a small village but then again, considering his age, he may not have had the Rune for very long.
I also find it's more than just a coincidence that not only does the True Rune of Punishment assault it's bearer with memories of previous owners and tries to goad the user into using it, but it also just seems a little too convenient that anytime someone seems to get the Rune, suddenly, a situation arises that calls for the rune to be used, which then promptly ends with their death. Perhaps it's just poor writing, but it's possible the rune itself simply alters the bearers fate to throw them into situations that require the rune to be used. I mean fate is kind of one of the underlying themes of the series...
Suikoden II flat out states the Rune of Beginnings forces the bearers of the Bright Shield Rune and Black Sword Rune to fight each other, so it's power to slightly affect fate or choose hosts with the correct criteria is pretty obvious.
The Elemental Runes are stated by Geddoe when he explains them to Hugo. Luc I believe gives a similar speech about the Runes desire to be used as well, though my memory of the game plot is fuzzy from not playing it for awhile. The Manga adaption elaborates far more into it and it's shown in the manga that they do in fact assault the bearers mind in order to have their power released. The True Fire Rune haunts Hugo's mind with the Flame Champion's memories and if Hugo gets too wound up in them, the True Fire Rune releases it's power. The biggest moment being in Volume 9-10, when Hugo is haunted by the Dharma Future that Luc shows him and the Rune nukes Hugo's room in Budehuc Castle. Geddoe makes a comment both in the manga and in the game if you choose him to be the FC that he has already seen the bleak future, so the True Runes have tried similar ploys. The Sun rune also seems to warp it's bearers mind unless the Dawn and Twilight Rune are there to calm it down.
While the idea of these runes being "cursed" is not as likely, it is still pretty obvious from some of the source material that none of the True Runes make their hosts life a pleasant one, if even the elemental runes give their owners a hard time. Even in Suikoden I, the people within the Dragon Knight Order flat out tell you that bearing the Dragon Rune is a curse for Joshua.
I'd say the runes are somewhat of a "curse" to their owners. While the Rune of Punishment is pretty much flat out cursed, and the Soul Eater is debatable. The other Runes are only cursed so far as to what the bearer considers to be a problem. I mean Suikoden III ends with Hugo, Chris and Geddoe all returning to their normal lives despite two of them now having immortality and the third has been doing a pretty good job with the immortality shtick before the game started, so I don't think the immense power and immortality is considered a problem. Sierra herself was living peacefully in a town she built with other vampires until Neclord stole the Blue Moon Rune and the entire Dragon Knight Order is based around following the bearer of the True Dragon Rune so I would argue the idea of the solitary life of being a True Rune Bearer is not really true. Only Ted, Tir, and the very long known list of Rune of Punishment bearers really follow the solitary/Doctor Who experience you suggest, and both Runes are the ones that are called cursed in the games.
I feel all this points to the idea that the True Runes are not simply "godhood in a can" er... rune, but that they are seriously forces that shouldn't be messed with and most people try to avoid. The games themselves show that bearing the rune is a heavy burden, not just because the runes are often associated with historical events but just having them garners problems of their own as the rune either has what appears to be a devious power or actually just messes with the users head to get its way. .
True beauty exists in things that last only for a moment...