can these LLP's raise money in the public markets--equity and debt?
Debt, why not? If it's been done by a UK LLP, I haven't seen it.
As for "Equity", I am sure that has not been tried, but of course proportional "Equity Shares" are exactly what I am suggesting could be an optimal new "asset class", with more in common with units in a Canadian Income Trust than anything else.
Inside an LLP is a very strange twilight zone in accounting terms: a Member has an account with an LLP, and "Equity" and "Debt" are essentially continuous. "Any economic unit can emit money. The serious problem is to get it accepted" Hyman Minsky