The Arduino Web Editor Chrome app subscription is significantly cheaper if you have an educational account:
https://forum.arduino.cc/index.php?topic=465962.msg3195056#msg3195056In the Education environment if your classroom is using Chrome OS devices, to get started please visit this link and subscribe to the plan that suits your school's needs, plans start at $0.20/student per month
That sounds quite reasonable to me but depending on your financial resources it may still be a consideration vs. the free Arduino IDE software.
Other than that, the major downside I see to Arduino Web Editor is that it's not so much under your control as the Arduino IDE software. If there is a problem with the server or the internet then your class is ruined. Updates will be rolled out with no notice and no matter how careful the developers are bugs do happen. With the Arduino IDE you might have some problems getting it installed but you can do that in advance and once it's on the computers and working you have a reasonable expectation it will work during the class just the same as when you were doing your prep.
A common issue we see reported here is caused by the >1400 3rd party libraries that are pre-installed on the Arduino Web Editor. Sometimes there are conflicts between the libraries. That would be much less likely with the Arduino IDE since you would only install the libraries you wanted and if there was a conflict you always have the option of removing the library causing the conflict. With the Arduino Web Editor there is no way for you to remove the pre-installed libraries. All those libraries are quite easy to install in the Arduino IDE's Library Manager (Sketch > Install Library > Manage Libraries) so I really don't see a huge benefit to having them preinstalled.
Another common complaint about the Arduino Web Editor is that it doesn't allow you to install support for 3rd party hardware. You only can use it with the official Arduino boards or unofficial boards that are compatible with the hardware definitions of the official boards. For an introductory class that would likely not be an issue since likely your students be using 3rd party hardware that requires the installation of a custom hardware package (ESP8266 for example).
There are some sort of neat "cloud" features of Arduino Web Editor which make it easy to share/collaborate/embed sketches. I haven't found any of that to be super compelling yet but I don't use it very often.
I know it is possible to use Linux on Chromebooks (Crouton) so that could be another option if the Chromebooks were the best value but you still wanted to use the standard IDE. I have no clue how feasible that would be for your usage.