.. not really
The amount of high level waste is not "tiny" in any financial sense. The current plans are for long term deep geological disposal, which are major engineering, expensive projects, relying on very complex safety cases for waste containment over timescales of 1000s years.
Reactors aren't just going to "come along" which " run on such fuel". The fast breeder reactor programmes were abandoned as unworkable. There is some scope for "nuclear incineration" of waste in PWRs, but not to the extent that they will eliminate all waste. Bear in mind that nuclear is a mature technology and is physics limited, with that physics being well understood. There is no equivalent of "Moore's Law" that's going to give automatic advances.
Meltdowns are not impossible with current new reactors ( i.e. those in production ). The industry has played games with the definition of passive safety / inherent safety. The EPR for example will still need positive action in the case of loss of coolant to prevent meltdown. Even if that action is automated and not reliant on off-site power, it could still fail.
SMR cost saving are all hypothetical at the moment.