Does anyone know anything about gardening?

Please help. My lawn, once a fairly lush green pleasant place to sit is now a barren wasteland most often resembling somewhere inhabited by wallowing animals. It has been on a downward trajectory ever since we had significant work done to our flat in early 2022 and they used the garden as general store and prep area. Due to various things, the garden never got the attention it needed to bring it back to life and the lawn was really finished off by the arrival of a massive puppy last September who was housebound for ~8 weeks whilst his jabs kicked in and he spent that time frolicking/trashing the garden. Generally, the lawn is shit, the ground rutted and uneven from the works and his intermittent digging. The dug-over section towards the back is where I have removed raised beds and turned the soil over getting rid of various weeds and escaped veg plants which were colonising. What I want to do is to try to regrow the lawn and turn the veg plots back to grass. I have bought a bag of mixed grass and clover seed. My assumption, from some brief reading, is as follows:

• flatten the dug-up earth (I have no suitable tools, but can probably do this with some large bits of wood lain over the patch and trodden down?)
  
• as best I can even out any dips/holes in the rest of the lawn and flatten this earth down
  
• leave for a couple weeks to see what weeds appear and pull them up (I won't actually have 2 weeks to leave it, but can manage 10 days)
  
• rake up a bit to loosen top soil and add some organic soil improver/light compost
  
• generously distribute seed per instructions
  
• water, if not forecast to rain
  
• keep puppy off the lawn for as long as possible (we are going away for 2.5 weeks after Easter, which is the window I'm looking at for lawn regrowth)
  
  

Is this a ridiculous idea? Am I better off not bothering with this and just getting some hardy turf down?
Anything I'm missing/getting massively wrong in this plan?

Also, any recommendations for hardy plants that might survive the attentions of an idiot puppy? I'm thinking grasses/ferns in the empty raised bed to the left, but might need to go established plants if they're to have any chance of survival.