I develop in Firefox and it’s pretty rare that I have to spend time on other browsers, which was not the case a decade ago.
It’s also not like Flash was perfect at this: I ran into compatibility issues where something worked on Windows but not macOS or vice versa, and the frictional cost of having an unstable, primitive IDE was significant — especially since Adobe typically ignored bug reports until the next major release, at which point they’d tell you that you should drop $500 for the privilege of seeing whether they’d fixed it.
It really depends on what you are developing. If you are using cutting edge browser features. Things like WASM or new JS APIs, sure you'll likely run into incompatibilities. However, if you are building a pretty standard content website, not a single page app, from my experience, pretty much everything just works between browsers. This was definitely not the case 10 years ago. Especially when including IE in the mix.
These days, it really comes down to “Do you need to support IE?” — even for fairly new features, if they're standard you should still check https://caniuse.com/ first but it's likely that the answer will be “Yes” for everything except IE11 and, far less frequently, Mobile Safari.
I always love new whizbang browser features. But I avoid them until they are common enough in the end user base that I can actually expect to be able to use those new features.
Now modern browsers that are self updating is a huge help.
It’s also not like Flash was perfect at this: I ran into compatibility issues where something worked on Windows but not macOS or vice versa, and the frictional cost of having an unstable, primitive IDE was significant — especially since Adobe typically ignored bug reports until the next major release, at which point they’d tell you that you should drop $500 for the privilege of seeing whether they’d fixed it.