Generally, illegal trading resolves around the idea of trading on non-public information. And front running falls under that category (access to order data that other participants do not).
However, HFT's do not trade on non-public information. Every participant has access to the same market data. I could start my own "HFT firm" tomorrow; I would just be incredibly unsuccessful at it because I don't have the finances or computing resources to execute.
This is a pretty widely discussed question, and while I think the empirical evidence is so far unclear, there are some obvious theoretical models where costs for large institutional investors (like pension funds) go up. E.g. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2238516.
However, HFT's do not trade on non-public information. Every participant has access to the same market data. I could start my own "HFT firm" tomorrow; I would just be incredibly unsuccessful at it because I don't have the finances or computing resources to execute.