But middle-class people often don't have 'generations of wealth and "class" behind them'. They could be lower-middle, one generation of being a rung above the WC and desperate to cling to whatever constitutes MC status for them. And yes, some LMC people are absolutely desperate to be 'better', in social class terms, than others.
It's why Hyacinth 'Bouquet' Bucket is a comedy monster, a crashing aspirational LMC snob trying to overcome her WC origins, her acquired accent buckling under pressure, referring to her modest house as 'The Residence', making her married surname more 'genteel', trying to give the impression she has domestic staff by answering the phone with 'The lady of the house speaking' etc etc.