It's your description that is completely wrong.
A transaction tax has been tried before in Sweden[1], had disastrous effects on their stock markets, and had to eventually be repealed.
I don't think most people appreciate how efficient the US markets are today, relative to what they were like 20 years ago. HFTs do make a lot of money, but its a extremely small amount compared to the total size of the financial system [2], and it is decreasing rapidly each year as the competition increases.
If we impose a transaction tax, market makers who are already on razor thing margins will increase bid-offer spreads, volatility will increase, and it will become way more expensive on average to trade. There's a reason why you can trade virtually for free on your personal account today, vanguard is able to offer you 0.1% in management fees, etc. These things did not exist 20 years ago.
Its a common myth that HFT's are "stealing" money from investors, but if these firms were not in the markets, it would almost certainly be detrimental to the ordinary saver/investor.
Bernie definitely wouldn't be able to raise nearly as much money as he would need from this tax, and it would hurt the economy. Him and his advisors know it, but he just wants to pander to the anti-wall street sentiment while also making it seem like his proposal isn't completely fiscally irresponsible, which it is.
[1] https://en.wikipedia.org/wiki/Swedish_financial_transaction_...
[2] https://www.ft.com/content/d81f96ea-d43c-11e7-a303-9060cb1e5...