If I was to make a homebrew barbarian subclass that allows barbarians to cast spells in rage, or allowed a barbarian that multiclassed into a spellcasting class, would it break anything? It doesn’t seem that it would, because the no spells while raging mechanic has always struck me as mainly mechanic, but I could be wrong.
It would break concentration by never breaking concentration.
This is the biggest concern. Rather than reinvent the wheel, I will echo Ruse's response to this question - Is this barbarian Rage Mage subclass balanced compared to the official barbarian subclasses and the eldritch knight?:
The fundamental problem with casting spells while raging is concentration spells. The Barbarian class has great incentives to have a high Con modifier and grants proficiency to Con saves, but what really sets it apart from other gishes (such as the Eldritch Knight) in this regard is the Rage.
Rage halves most of the incoming damage, so if the Barbarian can concentrate on a spell while raging, then breaking that concentration will be an order of magnitude more difficult. Moreover, Rage is an amazing buff and every other buff of this caliber requires concentration.
For these reasons, any subclass that lets a Barbarian concentrate on spells while raging is fundamentally unbalanced. It's not the kind of thing that you can balance out by making other features weaker.
A raging barbarian, especially if they take War Caster, which they will, will almost never lose concentration. Ever.
Possibly not, but you would likely need to playtest it
I am not averse to homebrew rules, but this seems to be changing a pretty important aspect of rage and I am not sure if the rule exists for flavour reasons or balance reasons.
My suspicion is that the rule only exists for flavour reasons, so it is not likely to cause much of an issue beyond possibly verisimilitude.
As pointed out by Thomas Markov there is a potential that rage would mean concentration is too difficult to break, which would be a very powerful thing, but in my experience a barbarian usually has low AC so would be making so many concentration saves it would eventually fail. There will come a time when the CON save is already +10, but by then some of the enemies will be doing huge damage so it should even out.
Even if it does turn out that concentration is difficult to break it is worth playtesting this on combination with a high AC eldritch knight, paladin or cleric (My mid level paladin has 24 AC using shield of faith and some items, and can still cast shield so is very difficult to hit, plus has crazy saves to often takes half damage from spells which is about what rage does. Plus his CON save is +8 which is about barbarian levels), and a well played druid, wizard or sorcerer (I say well played because that means using control spells to prevent incoming damage, and keeping out of lines of sight so you don't get hit) and comparing how often it breaks on each class.
Whatever your leaning, playtesting to see how house rules work is a great idea.