A necromancer who raises undead only at need, and disposes of them when their purpose is served, does not have to be evil.
If he does so often, yes he
does have to be evil. Says so in the PHB - '
ONLY evil casters do so regularly'.
That's not open to debate. If you regularly animate the dead, you're evil. Says so in the PHB. Animating them is
never a Good act (no matter why or for what reason you do so).
Whether that's a metaphysical consequence of using 'dark, foul unholy' magic (as described in the MM under the Zombie and Skeleton entries) to animate them (same as how a Jedi using the Dark Side in Star Wars gets corrupted), or a philosophical consequence of the choice to use 'black magic' to animate and bring into existence evil baby eating monsters powered by unholy dark energy, by defiling the corpse of a deceased sentient creature, is up for debate.
Barring DM intervention to say animating the dead is not, in fact, evil, you cant be someone who regularly animates the dead in 5E and be Good aligned.
Congratulations, you are quoting a rules straight jacket, akin to Paladin alignment restrictions of old. Yet, Eberron, and the world of the Elder Scrolls games both model entirely viable worlds, that differ from the default assumption in the PHB.
Each table of gamers gets to decide what mechanical rules, what flavor text, and what modifications to the aforementioned, that they wish to use.
Respectfully, the text quoted, may be RAW for you, while another group, may be fine exploring Necromancy as a dangerous tool.
I have no problem with this. A DM is free to ignore that passage of the PHB if he wants to, just like he can ignore any rule or text in any of the books he wants to.
But in 5E, by default, animating the dead with magic is not a good act, and ONLY evil creatures do so frequently.
Maybe the OP's DM runs things differently, I dont know. I'm just asserting the core assumption of 5E's necromancy and position on the animating of the dead (its eeevil) under RAW. If that differs in the OPs campaign so be it.
I'm very much a let-'em'-fight DM as long as it stays in character.
I'm very much a 'no way you can introduce that character into this party, without the informed consent of every single other player.'
Whether that's an evil PC being introduced into a good aligned party, or a good aligned PC being introduced into an evil aligned party, 2 clerics of opposing and hostile deities or any other similar scenario.
I'd only allow it if the players were OK with the high potential for PvP conflict, and (in my view) were mature enough to deal with it. All of the players. Otherwise I would veto the character, and require something more compatible with the other characters.