Why didn't Billy Russo delete records of the Anvil mercenaries

Anvil is a legitimate company. They do government contracts, including the security for Senator Ori. These people are not mercenaries, they are employees (they just so happen to also act as mercenaries, but officially speaking do not do so in Anvil's name).

If someone works in McDonald's, but also burgles houses at night, that doesn't mean that McDonald's is a crime syndicate. Similarly, the criminal exploits of Anvil employees are not necessarily related to Anvil itself. Russo's main argument is that the type of people who work at Anvil, are also the type of people who are likely to find other jobs as a hired gun.

This means that the government is aware of who works for Anvil, since they file their tax returns as a legitimate company. Anvil can't just make people disappear, it would raise questions. Especially if those missing files happen to coincide with missing person reports being filed (by the loved ones of the deceased Anvil employee).

The missing files would send up red flags to any investigator, as it indirectly proves that Anvil is aware that something has happened to this person. Even if Anvil didn't cause their death, it still shows that Anvil knows more than what they're claiming to know.

Russo doesn't even need to delete files, as he already have a (seemingly) valid excuse. When Madani first confronts him about the mercenaries being Anvil employees, Russo confidently responds that his employees are the type of people who would moonlight other jobs (which Anvil is supposedly not aware of), and that most employees are transient to begin with, they rarely stay employed by Anvil for a long time; and are therefore likely to find other jobs, supposedly unrelated to Anvil.

Not to forget he shot one of his partners in the head while fleeing from the fight in order to preserve his identity.

This was not done to hide the dead man's identity. This was done to leave no witnesses who would point the finger at Russo, or testify that Russo hired them.
Russo's claim that the mercenaries were acting on their own (not in function of Anvil) falls apart if people testify that they were hired by Russo/Anvil, or that Russo was present during the illegal mission.

Even if the witnesses have no proof; if several independent witnesses corroborate being hired by Anvil, that is enough for Anvil to lose its income (government contracts), even if they somehow manage to evade prosecution (which I doubt they would).

Not to forget he shot one of his partners in the head while fleeing from the fight in order to preserve his identity.

That's exactly why he wouldn't delete records. Looking at and stealing data is very useful, but has risks. Any time you access a system, good cyber-security will take note of it. But, still, with the distributed nature of the Internet, the cat and mouse game is concealing or covering your tracks.

To actually delete a record or files is something that no one, ever, will miss. To do so would massively increase the likelihood of leaving "fingerprints" that security experts would eventually track to you.

The other main reason is that it would be completely useless to do that. Any system is going to have regular datafile backups. Unlike when a nefarious evil-doer wants to erase Sandra Bullock from official records, and they don't want her information restored, if you delete some of my files, against my will, I'll simply go to the most recent data backup that has those files on record and restore them to my system. You've just exposed yourself for no particular advantage.