Not sure what area you work in but it isn’t the DWP that’s for sure. There is no yearly average !
Taking OP at face value (and I have 30 years in DWP) the situation is this.
savings of 16k or more knocks you out of UC or in fact any means tested benefits such as income support, income related employment support allowance and income based JSA. If you have been working and have paid sufficient NI in the last two relevant Taxi years (April 2024 to April 2026 then you could be entitled to benefits based on having paid into the system and these are not related to your savings however they are mostly time restricted months only.
As far as universal credit is concerned, there will be no problem with you purchasing a property as this is a long-term saving for the state. We do not pay mortgages until the nine month mark and then only the interest on them whereas we pay rent from day one. This is one of the greatest expenses in the benefit system as Rent is so astronomical.
When it comes to your remaining 60,000 in capital, deprivation of capital will only be considered if you have done something considered to be unreasonable with the money. For example, if you are to take your three children on holiday to Spain for a week, that would not be regarded as unreasonable. It is something that is considered fairly normal. If however you had taken your three kids and your parents on a world cruise for 6 months, that would be regarded as a deliberate deprivation of capital as it is not the norm and at the extravagant end of behaviour.
The type of thing that is normally accepted as not depriving oneself of capital are home improvements just keep all your receipts.
To be honest the example given about no bathroom or just one that you fancy changing is neither here no there. DWP does not have the staff or the inclination to become involved in your building projects so it matters not if if it’s a new bathroom for one you don’t like or just a bathroom because you don’t have one.