I am no lawyer, just a generic judgment, but...
The abuse comes because old law could not foresee new tech, easy travel, etc, and the scale of possible abuse. But this is abuse, not use, and it might be solved with an interpretation keeping the intent (the law cites the 'allegiance' concept) but not the explicit letter of the law.
About averse possession (usucaption), it usually requires a large period of time, otherwise owners would have to keep perpetual watch over every piece of land. The presumption is that legal property is valid, and the burden of proof is on the squatter, who must show he lived *undisturbed* on the land *for many years*.
Now, suppose a baby is born. If the parents have been living in the country legally for a long time, it would make sense to presume their allegiance and grant birth citizenship. But, what if the parents had arrived during pregnancy? Wouldn't it make sense to presume a temporary stay unless the parents prove they really moved their jobs, $$, etc, to the new country and are already in the path towards citizenship for themselves?
On another level, at that time USA (and other new american countries) relied on european imigration, so birth citizenship was a no brainer against european-like blood-based citizenship. Today the context is very different, but it would make sense to require a constitutional ammendment to change the letter of the law to a different principle.
A different problem is children of illegals. It does not make sense to grant citizenship to the baby and then deport the parents. And leting the parents stay because of the baby-citizen would make "babymaking" an automatic & safe citizenship path. Dangerous on many levels, including prostitution and fake or forced marriages.