I presume the real reason why children are omitted is due to cost. The venue will often have a maximum number of people (does this include children?). And if you are having a sit-down meal, then catering to children can be expensive, and youngish children will often not want to sit through several courses, and speeches.
When it comes to the actual ceremony, then a register office may also have a capacity, but the ceremony is fairly short duration. Similar to many other non-church venues.
A church wedding may have capacity, but usually the ceremony is a bit longer. I would expect children over the age of about 5 to behave themselves and sit quietly. But youngish children may easily find this difficult. Therefore, if invited, parents should be sensitive to this, and one parent should take the child outside if he or she become too noisy. I have seen some churches with a 'play area' at the back of the church, which might work, but it depends on the noise and level of disruption.
Weddings often have an 'evening' disco or similar after the actual meal, and it should not really matter if young children are running about, but OTH, many parents would not want their children out too late.
So, it really depends on what sort of wedding, and / or wedding reception is planned. Of course, children can be omitted, but similarly the B & G have to understand that this means some guests would have to refuse.