TL;DR: It's not impossible - but it's unlikely.
The Punisher had a 5-issue mini-series published in 1986, before Watchmen or The Dark Knight Returns. His continuing series launched about a year after the mini-series concluded, making it more likely that the success of the mini-series lead to the continuing title.
The Punisher first appeared in The Amazing Spider-Man #129 (cover dated February 1974). The inspiration for the character was Don Pendleton's Executioner, who appeared in a series of novels around this time. This was also around the time of the original Death Wish movie (it came out in 1974; the novel it was based on was published in 1972). This sort of character was not unique at that time, in other words.
As noted in the original question, the character appeared as a guest-star in Marvel's comics for years. Originally treated as a part of Spider-Man's cast about once a year through the rest of the 1970s, then started appearing in other character's comics (such as Daredevil and Captain America). He appeared in at least one or two solo stories, in Marvel's black and white magazines (which allowed freer reign for violence and such, as they were not submitted to the Comics Code Authority).
His 5-issue mini-series in 1986 was his first solo outing in the comics. It was well-received (actually toning down the character slightly, explaining away some of his more extreme actions), and his on-going comic starting about a year after the end of the mini-series.
Note that the first issue of his mini-series (given the cover date) would actually have hit the newsstands in 1984, so this predated both Watchmen and The Dark Knight Returns. In fact, his on-going started before Watchmen's 12-issue run finished.
While it was produced around the same time as the other two books, it also occurred around the same time as a potentially more relevant event; the vigilante shooting of four young men by Bernie Goetz in December of 1984.
Note that The Dark Knight Returns takes place in a dystopic future, and Watchmen takes place in the present, but one affected by decades of super-powered people. While Punisher certainly took place in the Marvel Universe, it rarely featured the kind of super-powered characters seen in the Marvel movies, or even in most of the Netflix shows (Luke Cage, Jessica Jones, Iron Fist). Frank Castle's focus was generally on the mob (as they're the ones responsible for the deaths of his wife and children) and street-level crime, not on super-villains.
Note also that comics take a certain amount of time between the decision to publish a new series and when the actual first issue hits the stands. On multiple occasions, Marvel and DC have published comic books featuring similar characters at almost the same time (a team of misfit heroes shunned by society lead by a older man in a wheelchair (X-Men, Doom Patrol), and swamp creatures (Man-Thing, Swamp Thing)). Sometimes, separate paths lead to similar results.
This seems more like the case with the Punisher. The success of the mini-series made it clear there was an interest in the character; other events of the time made it seem like society was ready for an on-going series about a character who was an ordinary man (OK, an ordinary ex-Marine) who was willing to take out the criminal element.
It's worth noting that other comics show the same thing. DC started publishing Suicide Squad in 1987, but laid the groundwork for the group in their Legends mini-series in 1986 (a year when they basically remade their entire universe). That group had no major compunctions with killing their opponents, either. Again, it's not impossible that the success of The Dark Knight Returns affected this, but (as stated earlier), given the lead time needed for comic book publishing, it would've been a quick decision.
A final aside: Marvel did publish comics where the lead characters were willing to kill before Punisher. While it wasn't as prominent or as obvious as the activity in the Punisher's book, S.H.I.E.L.D.'s stories about a spy agency did include killing their opponents. What's more, Marvel's "army" comics (like Sgt. Fury and his Howling Commandos, a predecessor of sorts to S.H.I.E.L.D.) certainly involved killing their opponents (as did similarly-themed comics published by DC). While I haven't read many of them, I imagine that Marvel's western characters (Two-Gun Kid, Rawhide Kid, Kid Colt Outlaw, etc.) at least killed on occasion.