This kind of question sounds similar to 'Am I the naughty word?' threads. Anyway, my answer to this is: Yes, you are too strict.
I generally don't like DM nerfing things with houserules, but if they have good reason for it, then fine, I will accept. I don't think this is the case with Wizard's ability to learn new spells. There's never a good reason to nerf it, and here's why.
This feature is 100% in the DM's power. No player would roll a wizard and go like, 'Now i can learn all the spells i want." First of all, the DM can control how much gold he gives out as treasure. Less gold available=less spells copied. Then there's the actual spell scrolls found as loot. DMs have total control over this as well. If you never give out a single spell scroll or spell book, a player can hardly complain, because it might be due to the circumstances or situation that the party is in.
The other part of the feature is the Arcana check to see if the spell is successfully copied. By RAW, a failed Arcana would mean that the spell is not copied AND the scroll is destroyed. This alone makes the feature unreliable, therefore needs no nerfing.
Its like, lets say you hate players having too many magic items, so you nerf the standard Attunement slots from 3 to 2. But you can just give out less magic items instead, or even have homebrew magic items with class requirements if you want to make sure that the Fighter isnt hogging all the magic items.
Another thing i dont like is DMs justifying their nerfs with 'but that doesn't make sense' argument. You say it doesn't make sense that the spell automatically appears in the spell book. But the same can be said for anything else. How does the Eldritch Knight suddenly have spells and spellcasting once he reach level 3? How can you apply Sneak attack when you are not sneaking? This is usually one of my red flags for bad DMing, that they hate a certain class and wants to screw you over for playing it. Even if they don't think so (every DM thinks they are fair), their houserules shows.
My point is, there are better ways to go about implementing your biases. We all have them, but its how they appear in the eyes of the players that matters.