They do - They're just not worth as much as American ones. The NBA deal is worth $7B a year? For the UK rights, The New deal announced by the premier league a couple months ago is a £1.8b ($2.2B)/year media deal. The thing is, even that is tied to performance - only 50% of the pot is evenly shared, 25% based on how often your matches are played on TV, and 25% of that pot is split depending on where in the ranking you land. So that means a team that doesn't get as much UK TV airtime, or do spectacularly well like Wolverhampton, might only get £70m while Man City gets £120m. The international rights are up for renewal next year, but are worth about the same as the UK rights, and I believe are evenly shared. But if you get relegated, you lose all of that revenue. - (there's parachute payments for the teams most recently relegated because of the expenses of running a premiership team, but it's 50% of the base UK amount, so at most they'll only get a quarter of what they used to - call it an average of £30m.) It's still a huge financial hit to get relegated, but equally expensive to run a competitive premiership team.
The EPL also kicks down to EFL solidarity payments from their media pot of £245m - but that gets divided by the remaining 64ish clubs so they get ~3.8m each. For context, EFL's own newly announced media deal is only worth ~£179m a year, and international rights only brought in £37m/year or ~$47m. That's for a 72 EFL clubs - there's a formula that breaks it down, as Championship obviously gets a bigger piece of the pie than League 2 - but that works out on average to about £7.2m per EFL club which just isn't a lot. Without the EPL solidarity money, on average it's just £3m, or $4m, and my understanding is that league 2 teams get as little as £900k. At the same time, NWSL's TV deal is worth $60m across 14 teams - they're getting more than $5m on average. The TV landscape is just different.