scholarships to private schools specifically aimed at state-educated pupils

Friends of ours were self employed, they owned a second home which was their future pension. Their actual income wasn't that great but when they applied for a bursary they were told that they had to decide which they want more a pension or independent education. This is fair IMO many on substantial bursaries have no assets and no decent pension.

As a few others have pointed out: if you have significant assets then you will be expected to free them up to pay for school fees.
Most scholarships are tokenistic in monetary value as schools prefer to offer bursaries to families that have very able children but can't afford full fees. Having an able child but lots of assets might mean you don't qualify for any means tested assistance. I have never heard of coming from state school being the only criteria to give a reduction in fees.
Th policy of only offering assistance to families who can't afford the full fees and don't have assets to free up is IMO a very fair policy. I wouldn't want to see schools offering bursaries aimed at low income families to families who have lots of savings, several houses, or a couple of brand new range rovers. Of course everybody is entitled to drive a reasonable car and own a house with a mortgage (which they occupy) and be entitled to assistance if their income is low enough.

OP if you do indeed have "too many assets" is there any reason why you might not free some of them up to cover fees?

You'll know already that most financial help has shifted from scholarships to bursaries - I doubt that many schools would hand over a significant bursary without means testing - mostly they simply use the scholarship exam to narrow the field of bursary applicants. (Writing slowly I realise this will already have been said ...)

As regards boarding - it's fabulous for children who enjoy it. But don't underestimate how big a change it can be. What does your DS say about this possibility?

You could look at Christs Hospital where I believe the fees you pay relate to your income/assets.

Having just reread your original post I thought I'd just add that if you've been refused a bursary before because you have "too many assets" I doubt you're going to be successful again, nearly all scholarships like the Beckwith are means tested I doubt any school will hand you a cheque for all of the fees without enquiring about your assets, and taking them into consideration. Although having said this boarding is obviously significantly more expensive than day so you may get some financial help but perhaps not as much as you need.

Rugby imply that they have bursary awards to children not being prepared for CE to me that indicates those in the state sector.

I know someone whose just got a big bursary out of Oundle admittedly at a prep but what have you got to loose by asking?

Abingdon have boarders and claim to have generous bursaries. Lots come form the state sector at yr 7.

Radley offer financial help to children in the "maintained sector" again you need to check deadlines etc.

Here "children of the clergy" I thought so.

Bit of a long shot but I think if you have ecclesiastical connections Marlborough I believe offer generous bursaries and will accept state educated children.

Bradfield offer scholarship/bursary to state sector children.
Harrow offer the Beckwith scholarship to children in the state sector they pay for a (?boarding) prep for yr 7 and 8 then a scholarship into Harrow. There was a programme about it I think it's on you tube. You might have missed the boat on that one or be about too.
What about Christ's? I suspect you're too late for yr 7 entry but they also have a yr 9 entry.
Colet Court also take state educated children from yr 7 then a bursary into SPS again you might have missed the boat I'm afraid.
If you DS is bright and going onto a good state school could you not tutor him at home to bring him up to the required standard for a scholarship into a good school for these don't you don't register till yr 8.

Ardingly have a scheme supporting children from local state primary schools. Not sure how much further afield it goes.

The Eton New Foundation scholarships (for state school boys) have a totally separate procedure and timelines to their main admissions, and you need to get in touch with their access adviser if you want to look into this. They only make a couple of these awards a year, and the academic standard is very high. But if you have a clever boy, there's no reason not to give it at try.

If you want London area boarding, Harrow pretests in year 7 and has various types of awards (including some with quirky conditions) so perhaps worth looking into, even though it's not a specific scheme for joiners from the state sector. Dulwich College might be worth a look too. They are both day and boarding (though boarding community small before 6th form), have quite generous awards, and have their own entrance exams which aren't that closely aligned to the CE syllabus.

But it's important to get the school in which your DS is likely to flourish, and which works for you logistically. Which parts of the country are you interested in? Boarding/day? Single sex/coed? Any particular things your DS that your DS is interested in?

This is probably a bit far ahead for you, but I'm posting it in case anyone comes across this thread who can benefit from it.

The Ogden Trust offer scholarships for state school pupils to do their A levels in private schools if they are interested in a career in physics.

No idea about scholarships but if your son is currently in Year 6 then you've already missed the boat for 13+ entry to most private schools. Registration deadlines for most of them are in the autumn of Year 6, some earlier than that. Not sure about Eton but I think I read somewhere that the boy has to be registered before the age of 10 years and 6 months, so unless his birthday is at the very end of August I think you're too late.

Most private secondaries around here (London) pre-test and interview children during Year 6. DS is at a prep school and most of his year group have their offers by now (with the exception of those of us waiting for Westminster, St Pauls and a handful of boarding schools - don't know anyone who applied to Eton though so not sure how they do it).