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A mortgage is special when buying a property though as your lender takes out a charge (is that the correct term?) on the property which is recorded by the land registry. You could make rules that says each individual can only have one such charge which prevents second home purchase with a mortgage. It doesn't stop borrowing money another way though.
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The existence of LLPs and more importantly SLPs (as well as offshore trusts) predicates against what you've discussed.
https://theferret.scot/edinburgh-offices-linked-secretive-partnerships/
Well no, it's a loan secured against a portfolio in my specific example. But in principle it's just the same. Rich people will use debt very often as it's much more efficient. One other benefit of my example is that by borrowing the money instead of selling investments you avoid any CGT