All job applications in USA require candidates to mention if they need the company to sponsor a visa now or in the future (i.e sponsorship). You need sponsorship if you have short term visas like OPT, H1B etc. On the other hand, you don't need sponsorship if you have long term work privileges like US citizenship, green card/GC or similar.
Unfortunately, quite likely that companies in the US break the law by willfully rejecting suitable candidates who don't need sponsorship (examples below). There are other tricks to reject citizens/gc, for example, by posting job ads in obscure local news papers which no one reads, paying advertising companies to post job ads but not actually posting the ads etc. Not all companies do this, but some do because they know they can pay H1B workers far less than the market wages. On the other hand, some companies simply replace their workers with H1B workers (examples below).
Hence, I wonder if Citizens/Green card holders could improve their chances of at least getting an interview by simply lying about their visa on job applications of SOME companies (not all), i.e. by saying that they will need sponsorship. The companies where you could lie might have small revenues, prone to offshoring, had recent layoffs, i.e. are more likely to commit visa fraud.
What are the consequences of telling such a lie ? Upon selection, the company might assert they can reject you simply because you lied about visa. IMO, this is not a material lie, i.e. no one is actually harmed by this lie. On the contrary, the company actually benefits because they don't have to worry about visa restrictions or expiry for a long time. But, the downside is that they are forced to pay market level wages.
Examples of fraud in H1B hiring :
1 - united states v Marijan Cvjeticanin - Google this. Its basically a case of paying companies to post job ads which offer h1b, but jobs were not actually posted. https://www.justice.gov/sites/default/files/usao-nj/legacy/2013/11/29/Cvjeticanin%2C%20Marijan%20Complaint.pdf
2 - Southern california edison - https://www.theatlantic.com/business/archive/2016/12/fixing-h-1b-visa-loophole/509639/
3 - Disney - https://www.nytimes.com/2015/06/04/us/last-task-after-layoff-at-disney-train-foreign-replacements.html The American workers lost this case though - https://www.orlandosentinel.com/business/tourism/os-bz-disney-lawsuit-workers-20180508-story.html
UPDATE : Why would someone want to work for a company which they think might be abusing H1B visas ? I can think of some reasons for doing this.
1) If the employer is desperate and your expected salary is just low enough, then they might keep you. Having a job during this pandemic is quite important. You can always switch once the pandemic is over.
2) Say that you get an offer, get rejected only due to lying about your visa and then the job is given to someone on a H1B (you can dig to find out). Then, you can take action against the company.
3) Free interview practice.