The Biometric Residency Permits which my partner and I possesed had "No Access to State Funds" printed on them.
Your ILR BRP would not have had "No Recourse to Public Funds" on it; known as the NRPF condition; because those granted ILR can have Public Funds, if their ILR is valid.
What exactly are "state funds".
Carer's benefit? Care Home support? Winter fuel help? Council tax relief? EV purchase subsidy?
Yes. Yes. Yes (if you mean Winter Fuel Allowance). Yes for the welfare benefit Council Tax Reduction but no for the 25% single person council tax discount. No (unless it states who can have this).
"Public Funds" are most welfare benefits, social housing, homelessness assistance, social care.
Contribution based benefits are not Public Funds. e.g. New Style JSA, New Style ESA etc