The +1 AC bonus is the only important part and is obviously a benefit
The most significant part of the Integrated Protection feature is the +1 bonus to AC. A flat, stackable AC bonus is a quite powerful racial feature; it will always apply to your AC (unlike most other racial AC features that are overriden by armour) and your AC is extremely important in gameplay, since it matters every time you're targeted by an attack, and that will happen very frequently. The Detect Balance spreadsheet (a popular fan resource for those evaluating homebrew race balance) assigns a score of 8 points to a +1 AC bonus trait, which it considers to be "an unusually powerful feature", equivalent to a +2 ASI or advantage on a commonly used roll.
However, the other lines of the Integrated Protection trait are basically what are known as a "ribbon" in the commonly used design parlance. They are mostly irrelevant, since it will almost never matter how quickly you can put your armour on (especially for Warforged, who also don't really ever need to take their armour off once they're wearing it), and it would be extremely rare that someone tries to remove your armour against your will. These qualities might become relevant in some extremely specific circumstances, but the overwhelming majority of the time they simply don't matter. Thus they have almost no effect on gameplay, and can be removed without really affecting balance; they're a "ribbon" in the sense that they're basically just there for decoration.
If you were to just drop the armour-don/doffing mechanics from the trait and leave it as a flat +1 AC bonus, there would be no practical impact on the Warforged's balance (though it does make it a little less interesting as a race). However, if you remove the whole trait including the AC bonus, you should replace it with some other benefit to make up for it, since that AC bonus is very good.
Xanathar's Guide to Everything does have an optional rule about resting which suggests that if you sleep in medium or heavy armour as part of a long rest that should reduce the rest's effectiveness; if you use this optional rule, then the question of how quickly one can don and doff armour might be more significant in gameplay. However, an eight-hour long rest requires only six hours of sleep, leaving an hour on either side available to doff and don the armour (the trait specifically notes that Warforged can rest while doing this), so it usually would not matter.
Moreover, the optional rule specifically states that sleeping in armour is an issue, but warforged do not sleep; as per their Sentry's Rest feature, they can just stand conscious but inactive for six hours, so the rule strictly doesn't apply to them anyway. This optional rule does slightly improve the relative benefit of the Sentry's Rest trait, since it means the warforged can always be happily wearing their armour if their resting party is ambushed - on top of being awake and able to respond immediately. However, this is the case regardless of whether or not you are using the Integrated Protection rules, and has no impact on the balance of that trait.
