I don’t like this rule at all. Definitely among one of my least favorite rules in 5e. There are several things wrong with this rule. First, the stated reason why this rule exists is not balance, but it exists to make sure that a spellcasters turn isn’t taking too long, by limiting them to only one ‘noodly’ spell per turn to stop them from flipping through the books trying to find the two perfect spells per turn, rather than just one (cantrips are easier to remember and use, I suppose). Unfortunately it fails at this in my opinion because of reason number two: the placement in the book. The rule is listed under the “bonus action spells” header in the spellcasting section. This is right between the “action spells” and “reaction spells” sections, and both of those just say “You can casts a spell with an action/reaction” and have no real rules. So people basically glance over it and assume there’s nothing important there. This means that new players (thepeople who will take a ton of time on their turns if they have to find two spells) don’t know this rule exist. The people who do know about this rule don’t need it, because they already know what spells they want to use and are much faster at taking their turns (hopefully). Also, the fluff is entierly nonsensical “Because bonus actions spells are espescially swift, you [can’t cast other spells on the same turn]”, what? Wouldn’t it make more sense that swift spells would leave you with more time to cast other spells?
Honestly, it’s even worse than that, because once you know the rule it actually causes the game to slow down because of how noodly it is. When you are casting a spell you stop and think “Wait a minute, is this allowed according to the bonus action casting rule?”, and then you have to find that out (hopefully not on your turn, but it causes you to have to look up this rule more that you really should have to look up any rule). If I am DMing I really don’t care about my players following this rule, but if I am playing I will always follow it to the letter (unless the DM says otherwise, of course), because I have had to look this rule up so many times I can now quote it verbatim from memory.
I think that this rule could be ok, but it needs some changes. First it needs to have it’s own section in the rules book “Casting multiple spells in a turn”, or something. Don’t hide it among stuff people skip over. Second, it should probably be changed to just “Because the casting of spells is a taxing affair, you can not cast more than one spell of first level or higher in the same turn.”. This is how most people think the rule works anyway, the fluff makes a ton more sense, it is simple enough that you don’t have to look it up constantly, and as a bonus it finally would answer all those people who are very confused about how you can cast counterspell in the middle of casting your other spell (you wouldn’t be able to, because that would be two leveled spells in a turn, except it you are counter-counterspelling to save your cantrip, I suppose. But that’s a very strange edge case.)