I don't think it is hysteria.
IME even the best deodoriser in the world is just temporary, at first. Within a couple of weeks, even without fresh pee, the sun warms up and the smell returns. It's disheartening and the OP doesn't want to spend money on solutions that don't work.
Apply it once or twice a month for long enough and it'll probably eventually do the job: or at least do it enough that any lingering smell is mild enough to live with. So perserverence will be the key.
All commercial deodorisers cause wider wildlife issues so, if that's something the OP cares about, that's a factor to consider.
However, pee is a particularly potent fertiliser so any living plant will utilise the nitrogen, produce lush foliage because of it and permantly neutralise the smell. It's an option. Depending on the quality of turf, it might also solve the problem of turf burn as some of the older/cheaper astroturf can get very hot in full summer sun. As the OP is new to the house, this might also be something to be aware/cautious of.
I agree taking it up now is not ideal, if that's OP's choice. The new seed/turf will require moisture and so is better done in autumn (ideally) or spring (second choice) so it may be a bit of a stinky summer till then.