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I (and other people) scale Aid Another by the DC you beat, so DC10 = +2, DC20 = +3, DC30 = +4, etc. Which gives something like the same effect you have, but less on/off, and also allows more people to give meaningful help.
Advantage is a very non-linear thing, and averages out to +3.35. But it's effectively +5 if you need a 10 and +1 if you need a 20.
The other way to use multiple skill rolls is to have multiple effects; the wizard in the above example could have come up with a useful fact the bard didn't get. In a Diplomacy situation the bard might persuade the duchess but the paladin impresses her general.
Actually, if you have a bard with a +30 bonus the second character is irrelevant. The bard by that point has lore master and can take 10 on any knowledge check and can take 20 at least once a day. Your chance of making the check is 100% with just the bard. The Wizard is always going to feel cheated when a bard like that shows up.
Most of the bards I have seen don’t really have that high of a bonus on knowledge skills. They are CHA based not INT so while they get more skill points from their class than a wizards INT bonus usually means they are getting about the same number of skill points. The bard also has other skills they need to invest in. Versatile performance means the bard is more likely to max out performance skills than knowledge skills. They also have social skills that will be a higher priority to max out instead of knowledges. It is not unusual for a bard to put a single skill point into knowledge skills and leave it at that. What really makes bards good at knowledge skills is Lore Master, not massive bonuses.
The Wizard by level 10 will probably have a +19 on his knowledge Arcana, where the bard is probably going to be around +9. As they level up the Wizards bonus is going to grow faster. His INT will be going up and is more likely to max out the skill. At this point the only way the check is being made is for the bard to aid the wizard. Under that scenario there is a 10% chance to make the check. Assuming a 12 INT the earliest a bard can get a +30 bonus is 18th level and only if they have 17 points invested. Assuming a 24 INT the wizard at this level will have a +28 or higher bonus. More than likely the bard will probably have around +14 or higher bonus instead of +30.
While in theory your situation could come up the chances of it actually doing so are about the same as winning the lottery.
The real benefit from aid another is that it allows a group effort to succeed where the individual may fail. Since aid another only requires a DC 10 check any character can make that, even a character not trained in the skill. That means that every character in the party can attempt to aid another to give the primary character a +2. Unless they have a penalty to INT every character in the party has a 55% chance to give a +2 bonus. If a character has an INT bonus and or any points in the skill the chance goes up.
Under your house rules the only character in the party that can give a bonus is the wizard, and the bonus is going to be limited to +2. Under the standard rules every character in the party can attempt to help. Assuming a party of 4 the bonus would cap out at +6 instead of +2. The standard rules allow the entire party to try and help, your only allows a single character.
Yeah, the Bard in either scenario doesn't need anyone's aid to make that check. And, from my experience, very [VERY] few players actually use Aid Another.
Aid Another is awesome and wonderful, and doesn't need to be made anymore difficult to use than it already is... which it isn't [difficult to use], but people rarely use it already. We should encourage it's use, not discourage it. Then you get to see some incredible shenanigans, like Helpful Halfling builds handing out +7 [or more] to AC or attack or skill checks... possibly as a swift action, or even reactively as an immediate action/AoO. When you're casually tossing around Aid Another bonuses of 5+, like it ain't no thang, it gives literally everyone at least a small chance of succeeding at pretty much anything they set out to accomplish.
We all get by with a little help from our friends...
As a Bard, it is my job to inspire competence in my allies, and as such I am generally built for Aid Another. I have all the skills, so I can offer aid to anyone for everything all of the time. And my gloves add my Arcane Strike bonus to the bonus given when I use Aid Another. So you try first. I'll give it a shot if you don't get it, but I think you can do this...
The class of the secondary character does not matter. Unless the secondary characters class is getting some bonus similar to the bard (which a wizard is not) all classes have the same potential bonus. The only thing that makes a difference is if the skill in question is a class skill, and the bonus from the stat. You could use a fighter with a trait to make the skill a class skill and it would not matter except the fighter probably would not have as high of an INT so the bonus would be lower.
As I said the real value of aid another is that it can raise the maximum the bard can get. Each character that can make the aid another check raises the maximum the bard can achieve. This allows a lower-level bard to be able to hit the target number. If the bard is not maxing out his ranks in the knowledge skill the wizard will often be able to achieve a higher number but has a chance to fail. With Lore Master the bard is often a more reliable way to achieve the result you need. When the bard has a +9 and can take 20 that means he is 100% guaranteed to be able to make roll with a DC of 29 or less. The wizard with a +19 will be able to hit a DC of 29 55% of the time. His chance of hitting the DC drops 5% for each 1 point the DC raises to a 5% chance to make a DC 39. If the wizard aids the bard, the bard has a 100% chance of making a DC 31, where the wizard has a 35%.
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I (and other people) scale Aid Another by the DC you beat, so DC10 = +2, DC20 = +3, DC30 = +4, etc. Which gives something like the same effect you have, but less on/off, and also allows more people to give meaningful help.
Advantage is a very non-linear thing, and averages out to +3.35. But it's effectively +5 if you need a 10 and +1 if you need a 20.
The other way to use multiple skill rolls is to have multiple effects; the wizard in the above example could have come up with a useful fact the bard didn't get. In a Diplomacy situation the bard might persuade the duchess but the paladin impresses her general.
I use the scaling Aid Another house rule in all my home games, and it makes group checks a lot more entertaining, especially if you can Aid with a related check and speeds up play as the party chooses a primary to make the main roll, and the party then stacks their Aid bonuses on top.
If I want to limit the success cap I limit the amount of participants can "Aid Another".
I consider Pathfinder to be a team game and it sounds like the wizard's player has wasted resources by investing too heavily in knowledge (arcana). He would have helped the party more by investing in another skill.