People actually play with Yogg-Saron. I assume Yogg is over-represented on streams where there is a viewing public itching to see crazy things happen in games, but I can look up a Yogg-Saron rogue deck online and find miracle lists that appear to have eschewed better win conditions for whimsy of Yogg.
There's a lot more to Yogg-Saron than the average result. There's that feeling that Anything Could Happen. But the kind of anything that makes it talk about Anything is very unlikely to happen, and averages certainly tell us something about what to expect.
By restricting my card pool to spells, I came up with some handy numbers that might guide you in your evaluation of Yogg.
Net damage per spell to your opponent if you play Yogg on an empty board: 0.22
Net damage per spell to your opponent after Yogg has killed himself and the board is empty: 0.26
Net damage per spell to yourself: 0.038 with Yogg or 0.082 without
Chance that Yogg kills himself or otherwise removed himself per spell if he hasn't summoned any friends yet: 11.9%
Damage per spell after Yogg is dead and Embrace the Shadow has fired: 0.41 to your opponent, 0.35 to you
Damage per spell if your opponent is at 10 and we count Tree of Life: Just 0.13
Number of cards gained to your hand per spell: 0.2
Number of cards gained to your opponent's hand per spell: 0.015
Chances you'll discard your entire hand at some point along the way to your 15th spell: 5.8%
Armor gained per spell: 0.071146
Attack gained per spell, counting weapons: 0.071146
Fraction of spells that add the same amount of attack and armor: 0.0079
Average number of minions summoned per spell: 0.15 to 0.175 depending on how many murlocs have died this game
Fraction of those minions have have 3 or more health: 0.086
Chance for a minion with 2 health on your side to be killed per spell: 20.1%
Chance for all minions with 2 health or less on your side to by killed per spell: 3.2%
Number of spells that will put a minion into play on your opponent's side: Fewer than one
Number of spells that will remove a minion from your opponent's side, excluding by dealing damage: 12
Odds that your opponent ends up with a minion in play if they didn't start with one: Quite small
Over the very long run you should expect to have no minions in play more often than you have minions in play. Over the very long run, ignoring Embrace the Shadow, you should expect your opponent to lose about a third of a health per spell cast and for you to break even unless Tree of Life happens. You should also expect to draw a card for every five spells except when you have to discard your hand instead.
So if I tailor my entire deck around Yogg, cast 10 spells and then drop him onto an empty board then I should expect that Yogg won't make it though the process, that my opponent will take three damage and that I'll draw two cards. With 20 spells cast it's six damage and four cards, but with 20 spells cast there is a good chance those cards I drew are just hastening my death to fatigue.
When I first saw Yogg-Saron I thought he was probably worse than an unpumped C'Thun. I think that might be spot on. But if you do want to play Yogg, promise me one thing, don't play him and mash the End of Turn button. There is a reasonable chance you'll end up with attack, and even a chance you'll end up being able to cast something.