Give 'em enough rope, lose the safety net
This character is being enabled to fail by the DM, you, even with the best of intentions. You'd like for the player to enjoy a character concept that he's come up with. The problem is that his play style is suicidal, and as you note (1) his PC dies frequently, and (2) the rest of the players "get" team play far better than this player does.
You have tried coaching and counselling (Well done! I've discovered that coaching is a part of the DM role). At this point it is no longer your problem to solve: it is his problem, and - this is important - the party's problem to solve. The rest of the players have been carrying this player and his suicidal style.
Next time the PC dies, ask the surviving party members their plan
This next time, it looks like there will be no "repeated reincarnation".
I killed the very NPC who was responsible for bringing him back! And
then had a sidebar conversation about how death was real and needed to
be treated that way.
This puts the burdeon on the party as a whole to deal with a dead character.
Address them as characters and ask them:
What is your plan for getting {this PC} resurrected/raised/reincarnated?
If they don't have one, he gets to roll up / create another character whom they'll meet soon during the next encounter.
If they do have one (perhaps paying for that reincarnation spell) let the dynamic between the players - your problem player and the rest of the group - play out. For example: it may be that he needs to carry his own diamond for revivify (if your party has a cleric).
There are five players in the party, and they just reached level 5
At level 5, if the party cleric needs to keep a slot for revivify always prepared (thanks to this PC's suicidal style) I know that a condition I'd consider as a cleric player is something like this: "your character needs to always have a 300 GP diamond on his/her person. I'll need it to revivify you if you expect me to bail you out yet again."
Not sure how your other players view this, but that's one way to marry up the in-game and out-of-game communication at the table.
Choices have consequences
You mention that this player gets upset if they get incapacitated. That tells me that this player does not actually understand this edition's mechanics. Dropping to 0 HP and having an ally pop you back up (or getting that lucky death saving throw of a 20) is a part of this edition. (Some people call it yo-yo healing and don't care for it, but it's a part of the mechanics).
He has played D&D off and on for years, both 3.5 and 5e. He understands the mechanics thoroughly.
This player may be a slow learner, who also seems to not understand team tactics. This is not uncommon; I've seen that manifest in various ways over the years. If the character keeps choosing to behave as a fragile tank, the consequences are that the PC will often be dropped to 0 HP or die. The former is no big deal, with this edition's recovery mechanics. The latter is a bigger deal for both the player and the team.
Until the player accepts that, I don't think you'll see a change in behavior. In my experience across numerous editions, enabling play that does not fit what the team is doing is counterproductive in nearly every case.
Your last bit of advice to this player: be a better teammate
You can offer some general advice along the lines of
If you keep doing the same thing and expecting a different result, you
may want to consider changing what you are doing to get a different
result
The point of emphasis ought to be on teamwork.
You mention in a comment that
"His character regularly goes off on his own even during RP. So the rest of the party will be doing one thing, and he'll be doing something else entirely"
That's not teamwork, and it also an added burden on the DM. Make sure that you tell him that his behavior is making your job as DM more difficult. (One of a number of fine Q&A on dealing with split parties - there are a bunch of good inputs here on how to deal with this if it keeps coming up).
If he chooses to fit into the team approach to the adventures better, his PC will either stop dying or die less often. There's a lot of room for being 'down but not out' in this edition. Older editions were a lot more lethal in terms of "you are dead, make a new PC" than this edition.
Best Answer
Complete Divine (pages 129 - 130) is the 3.5 supplement that covers this. It suggests that, as a default, the following is true:
Again, this is just the base suggestion for the default setting, but is probably the closest to a definitive answer you'll find for 3.5 in general. Specific settings might have their own rules.