I'm having difficulty understanding how Necromancers (as in the Wizard subclass) can pull their weight at lower levels. Necromancy spells are few, vary in ranges (often require melee range), provide no defenses, and have very few support effects for Wizards.
The School of Necromancy benefits are fairly negligible, as they require you to land a finishing blow and have lost health, at a level where two attacks can off you without difficulty.
I cannot see how a Necromancer has any realistic use in lower levels compared to almost any other caster.
But perhaps I'm missing something. What value does a low level (2-4) Necromancer bring that another choice wouldn't surpass you at?
My goal is to see more diversity in my groups, and one thing preventing a lot of players from ever touching the School of Necromancy is its first 4 levels, and nobody wants to be a stick in the mud for 6 sessions. With a solid answer, I will set the early scenarios to make our group's necromancer feel as valuable as the team's bard, hopefully without bending too many rules.
In case it is relevant for any decision making, characters will be made via Point Buy, and I would expect the campaign to run until about level 7 (just kind of a vague guess at the moment).
Best Answer
As valuable as any Tier 1 wizard: wizards are generalists by design
The limit of the scope of this question is from levels 1-4. The wizard does not choose a school until level 2. A focus on levels 2-4 leads us to initial spell selection. That single feature, the six initial spells and three cantrips, selected before adventure day one, is step one in the Necromancer Wizard, or any wizard, being good for the party. The choices made for second level spells (a total of four of them at levels 3 and 4) are the next significant levels of being beneficial to the party.
There is no requirement to choose necromancy school spells initially, though choosing some is valid if the character is being built with an eye to the future. If you require that the character choose necromancy spells as they level up, as the DM, the problem isn't with the Necromancy school of wizardry. As a game design matter, levels 1-4 are not that crucial to character effectiveness (see PHB p. 15). A power and capability spike happens at level 5: multi-attacks for martial characters and 3rd level spells for casters. New features for all wizard sub classes arrive at level 6.
All of the party's characters at levels 1-4 (Tier 1)...
A core benefit at second level regarding necromancy school spells is that spells that are found cost half as much to put into the spell book -- as with each other school.
The other benefit, the occasional hit point gain for killing an enemy, takes pressure off of the healing resources of the whole party. Design paradigm: it's a team game.
At first level
The problem is that there are too many good spell choices. The quick build offers an OK starting point for any school of magic.
Experience base: six wizards played (Five, if the UA Loremaster is discounted, which IMO it should be).
I'd suggest firebolt over light depending on party make up, and chill touch over ray of frost since it has a chance to foil attacks by undead against the wizard casting it. (Also thematic for your necromancer). As I've not played with toll the dead I won't comment on it, but it looks to be a good fit.
You can pick up area of effect spells, like shatter at level 3, and more battlefield control spells (example: web). The point is to help the team succeed. That is true for any wizard. Both of my brother's wizards (illusionist and transmutation schools) focused on party support and various attempts a battle field control at low level since that was the best way to leverage spell casting (beyond the use of cantrips).
There is no need to worry about the necromancer
I don't find the concern to have much merit, unless the game will only last until level 4. From the comment you provided, the campaign is expected to reach level 7. The Necromancer has plenty of chances to shine as new spells and class features arrive.
They can theoretically find and place into their spell book every wizard spell in the game. No other arcane spell caster can do that.
How many additional spells the wizard has access to (beyond what they can pick at level increase) is up to you as the DM / adventure-loot quality control specialist. How many spells, spell books, or scrolls do you intend for them to discover at early levels? They are generalists as a class, regardless of magic school specialty.