No, there isn't likely to be a bromine shortage.
The US is a major producer of bromine.[1] It's not at all rare. It's just that the cheapest source is the Dead Sea, because that's concentrated brine. There are bromine wells in Arkansas. It's a by-product from some oil wells. It's in seawater. In California alone, the Salton Sea and the SF salt evaporator ponds are potential sources.
If the price goes up, the use of bromine for pool chemicals and fracking fluids will be affected long before the semiconductor industry.
[1] https://pubs.usgs.gov/periodicals/mcs2026/mcs2026-bromine.pd...
The problem is high-quality hydrogen bromide, from the article.
"Critically, ICL’s hydrogen bromide gas production, including the semiconductor-grade output supplied to South Korean fabrication plants, is manufactured at the same Sodom facility where extraction occurs, meaning extraction and conversion infrastructure are co-located in the same vulnerable corridor."
The US production in your linked article is listed as "W". This is explained as "Withheld to avoid disclosing company proprietary data". But imports consistently exceed exports, so it appears that US production is not likely to make up a global shortfall.
That's not what the article is talking about as a chokepoint, and it does describe US bromine production.
Dow Chemical operates brine wells from which it extracts bromine in the middle lower peninsula of Michigan as well. Around Mt. Pleasant, St. Louis, and Midland. Besides all the uses you listed, it's also widely used as a fire retardant.
In 1973, Velsicol Chemical Corporation, who was operating in St. Louis, Michigan at the time, was manufacturing Polybrominated biphenyl fire retardant, as well as animal feed supplements. They were bagged similarly, and PBBs were accidentally shipped into the food supply. Which led to the largest livestock culling in US history at the time. https://www.michigan.gov/mdhhs/safety-injury-prev/environmen...
> If the price goes up ...
That is what "choke" means in the global economy perspective. Even slight price increase on such material can cause inflation and that's everyone's problem.
Part of the real confusion people have is - so many things about current trade are due to "current economic decision making". That is, something isn't rare or unable to be done elsewhere but that it's been done this way for efficiency of all involved.
There's often a really weird undercurrent of nationalism that springs up in these dicussions as if its' "a country" that does something well as a function of being that country, not as a function of an economic opportunity and ramp up.