Thanks for your nice solution - “It's lovely to experience children('s) playing in the garden again.“, which, I think, means, "Children are playing in the garden again, and it's lovely to experience that event or situation.”
Yes, deepcosmos, that is the meaning, both of the paraphrase and of the original.
While I also considered that paraphrasing, I thought such a construction with gerund could not distinguish the sentence with simple bare infinitive form - “It's lovely to have children play in the garden again.“ from the original one.
And then, under the paraphrasing with ‘experience’ in the place of ‘have object do/doing’, is there no other way to distinguish above two construction due to the attribute of the verb – ‘experience’?
One of the reasons English has so many constructions is that each one tends to be special in some way and not one-hundred-percent replaceable by another.