Ahhh fuck 'em.
You know what? We took DS for a walk in his buggy the other night, as a treat - it's been in the shed for the last year and a half but before that was still getting regular use up to nursery etc. He's nearly 5!
Exercise doesn't have to mean shouting them along the street - there's kicking about a ball, soft play, trampolining, going to the park - all much better fun. There is no need to force a 3 year old to walk along the street when they don't want to and you have an easier alternative - pick your battles.
Re the tantrums, there's heading them off at the pass with diversions as a first recourse (like if it's I waaaannnnt thaaaat, ignore and derail rather than head-on collision iyswim). He's old enough for the naughty step, but read up on how to do it properly first, and bear in mind you'll need to apply it a few times (which can be stressful) before seeing results. That said, we did it and all I have to do now is start counting if the little guy begins to misbehave - if I get to 3 he knows he'll be taken to it.
For the sleep.... ummm... blackout blinds and a slightly later bedtime?
ANYWAY all your problems sound perfectly normal, and it's a classic case of your mother blaming you for not being a SAHM, which is bollocks. Do what I do when my parenting is criticised by my parents. Smile and nod, inwardly remind yourself of the myriad ways in which they cocked up, and do what YOU think is best.