Suggesting that Snape joined the Death Eaters because he was “friendzoned” by Lily Evans is a fundamental misreading of the Harry Potter series.
There was no pity in Lily’s voice. “It’s too late. I’ve made excuses for you for years. None of my friends can understand why I even talk to you. You and your precious little Death Eater friends – you see, you don’t even deny it! You don’t even deny that’s what you’re all aiming to be! You can’t wait to join You-Know-Who, can you?”
He opened his mouth, but closed it without speaking.
“I can’t pretend anymore. You’ve chosen your way, I’ve chosen mine.”
If Lily is correct in her assessment of the situation, then the breakup of their friendship had little bearing on his politics. The true tragedy of Snape was that he was already on his way to becoming a Death Eater whilst they were still friends.
Lily made excuses for his behaviour, evidently believing that he was making an out-of-character misstep with his beliefs, and that he’d eventually come to his senses. Eventually, he uses a slur against her directly, and she draws a line.
“You’ve chosen your way.”
There are many reasons why Snape may have been drawn to the Death Eaters, such as:
his unhappy upbringing, with parents in a mixed magical/Muggle marriage
his ‘doesn’t like anything much’ angry, domineering Muggle father
his potential friendships with his Slytherin housemates
in a world where the Slug Club existed, him quickly realising that networking was his only way out of living in poverty in the Muggle world
thus, potentially being groomed by older housemates, such as Lucius Malfoy - the “lapdog” reference by Sirius suggests strongly towards this
the animosity between Snape and the Marauders, who were openly advocating as being anti-Death Eater (the enemy of your enemy is your friend)
the apparent lack of reaction from Dumbledore when Snape was almost tricked into walking into his death, putting Snape at odds with Dumbledore - who was the leader of the opposition
arguably, the constant bullying wore him down to a point where he desired power and status as a method of making others finally admire him
Losing Lily’s friendship may have caused him to throw away any lingering doubts about joining - if she was the only reason why he didn’t fully commit to the cause, if she was no longer his friend, then perhaps he had no reason not to sign up.
…but it was not simply ‘because Lily spurned him’.
It’s also highly likely that once she started to date James, Snape would believe that her judgement was clouded - if she could date someone as awful (and make no mistake, no matter how James behaved towards others, the way he treated Snape was appalling) as James, then what else was she wrong about?
To Snape’s horror, he soon found out that he had been misled and that Lily had been correct all along. It is to Snape’s credit that he attempted to right his wrongs once the penny finally dropped with him.
When he met with Dumbledore, he was risking death for disloyalty (as we saw with Regulus and Karkaroff). Snape had already begged the Dark Lord to spare her, yet he evidently didn’t trust him to do so.
When Lily dies, Snape wishes that he was dead - I’d argue that he expected to die during that year, whether at the hand of Dumbledore for daring to meet him, or at the hand of the Dark Lord for daring to defect, or at any point along the way if he was discovered as being a spy and a traitor, but he truly believed that by putting his faith in Dumbledore, Lily would be saved.
But, as the world is cruel, it was Lily who would die instead whilst he lived on. With the guilt of her death heavy on his shoulders, Snape began a life of effective servitude, preparing to be thrust back into the war once it resumed, repeatedly atoning for his sins as a youth by his heroic actions in the war, saving the lives of all those he could.