Why does the lack of ATP in muscle cells cause the muscles to become rigid rather than limp soon after death?

Because ATP is used for relaxation of muscles, not the contraction.
It is indeed quite counterintuitive, because ATP is always associated with ‘action’. This is different for muscles, so lets first take a quick look at how muscles work.
muscles in resting state actively pump out calcium ions ##->## a membrane potential is created
a contraction signal of a neuron causes calcium channels in the muscle to open ##->## calcium flows into the muscle
there the calcium ions interact with myosin ##->## muscle contracts (myosin interacts with actin)
the muscle stays in contracted state until ##color(blue)”ATP”## binds to myosin ##->## relaxation of the muscle (myosin looses interaction with actin)
at the same time as step 4, ##color(blue)”ATP”## is also used to actively pump calcium ions back out of the cell ##->## restored resting potential
Knowing all this, the rigidity of muscles after death (rigor mortis) can easily be explained: When breathing and circulation stop, muscles become oxygen deprived and can’t generate ATP aerobically. They might switch momentarily to anaerobic respiration, but they will soon lack sufficient amounts of ATP.
Due to the lack of ATP, step 4 and 5 can’t be carried out: myosin is not released form actin i.e. the muscle stays contracted. In addition, the calcium ions can’t be pumped out of the cell, so resting potential can’t be restored.