I gave it some thought, and the weapon set that makes sense to me is the following - some of it won't sound right off the bat, and there's a little bit of a shakeup in there, so be patient and I'll explain the thought process behind it!
1: Shield Gun (starting weapon)
2: Link Gun (starting weapon)
3: Bio Rifle
4: Shock Rifle
5: Ripper
6: Minigun
7: Flak Cannon
8: Rocket Launcher
9: Lightning Gun
0: Redeemer
Shield Gun
The first weapon is a given - it must be a melee weapon that allows the user an increased degree of mobility within the game environment - it has historically always been the case and a movement based tool as a starting weapon is a good cornerstone to work from. I elected for the Shield Gun as historically it has broader functionality over the Impact Hammer - though that extra shield ability is a contentious one. Both the primary and secondary fire draw from the same ammunition pool, which simply slowly recharges over time. A notable departure from the UT2Kx variety, I believe both primary and secondary fire should be useable at the same time - though it comes with the exceptional cost of rapid drain on your ammo pool, and the inevitable wait before the weapon becomes valid again.
In a fairly rare design decision, the primary fire should incorporate all of the flexibility of it's predecessors, so it's less of a weapon and more of a tool with a fair amount of nuance - primary fire is the same charging , high damage attack we all know, but when fully charged it can knock power-ups from opponents and deflect incoming rockets. The shield component needs to change - the UT2Kx version allowed a player to be too defensive and artificially prolonged matches more than it should, particularly in the case of duels. Instead of a perfect shield, the player should be able to take a limited number of hits or a smaller number of hits from radius damage (damage value itself irrelevant) before the shield becomes useless - I'd propose a system where the shield has a number of visible tiles, each of which gets depleted in turn on impact, but will come back over time (consider a hexagonal formation, with a central hex tile and surrounding hex tiles as an example). Using the tiled system, the shield is largely redundant versus rapid fire weapons and weapons with a damage radius, but effective against single shot weapons, particularly the Lightning Gun.
Link Gun
This is part of the big shake-up. The Link Gun has always been a mid-slot weapon, and the starting weapon has otherwise always been the Enforcer or Assault Rifle. The trouble with the Enforcer and Assault Rifle is that their primary fire modes have always been redundant - the Minigun variations offer better rates of fire, and the Shock Rifles have offered better damage per shot and more flexibility. The Assault Rifle was only really useful for it's very limited pool of grenades, which in turn were made redundant by the Flak Cannon and Rocket Launcher. In both cases, the weapon was used best akimbo, where it became more dangerous - but this means killing another disadvantaged player in order to obtain it, and doesn't change the fact that the weapon isn't providing a role that isn't already maintained by other weapons in game.
Given that the weapon is essentially redundant and often ignored, why include it in the first place? I decided the Link Gun was the best option to move into this slot due to it's use and behavior. In regular deathmatch, it is still a basic rapid firing weapon at heart, though usually not favoured over other weapons due to lower reliability at range. At closer quarters it does become considerably more dangerous - which is what you surely want in a starting weapon, as it encourages a bit of aggression. Additionally, the weapon has good applications in team games thanks to the linking ability, allowing newly spawned and disadvantaged players to 'gang up' on better equipped aggressors. In certain team games (AS/VCTF/ONS/WAR) is has already been included in almost every spawn point weapon locker anyway due to the ability to heal vehicles, nodes and potentially objectives.
The Link Gun's functionality should basically remain unchanged - a rapid firing projectile weapon with fairly all-round stats, and a high damage short range beam that can heal certain objects. I'd like to see the UT99 link beam 'sawing' mechanic come back, as this encourages learning some technique for higher level play. I'm unsure exactly how the weapons should link, as I'm neither a fan of the UT2Kx variation nor the UT3 one, though I lean towards UT3.
Bio Rifle
No introduction needed, and I wouldn't propose much in the way of changes from it's UT3 counterpart. The only changes I would like to see are in the projectile behaviour; I'd like to see the big blob being broken up into smaller blobs if hit with weapon-fire in mid flight (as this just feels right to me), and for blobs to respond a little more to physics - for example being pulled and thrown around by the Shock Combo Explosion.
Shock Rifle
This weapon is intrinsic to UT and needs no changes - you cannot have UT without it.
Ripper
This is the other part of the shake-up - the Ripper hasn't made an appearance in UT outside of the original game but has remained a fan favourite for many (and is a source of ire for others). The core concept of the Ripper is in itself not a problem, and I see it is a perfectly viable new mid-slot weapon. The Ripper primary must have a fairly low rate of fire, but offer a moderately fast, linear moving projectile with a high damage output, but only on direct hit. The projectile will bounce once on contact with world geometry and be destroyed on subsequent contact. The headshot capability should remain - but it should not apply to bounced projectiles which should apply around two-thirds the damage of a direct hit.
Minigun
It's had a few names, but the Minigun is exactly what we know it to be. I'm considering that this weapon should be simple in behaviour, contrasting with other weapons, and have no alternative fire mode.
Flak Cannon
Another weapon needing little or no adaptation.
Rocket Launcher
The three-barreled variant from UT3 and UT2Kx is the direction to take, and should probably incorporate all features from it's predecessor. The grenade launching feature may want some additional work to make it more viable in combat, as it was fairly unused in UT3 save the occasional chipping up onto a low ledge.
Lightning Gun
This is one of those divisive ones, but as I see it the Sniper Rifle and Lightning Gun are largely functionally identical, so there is little to debate in that regard. UT99s sweat by the Sniper Rifle, but there is really no reason to other than nostalgia that I can see. The Lightning Gun's design and implementation was simply more 'Unreal' - twitching corpses and flaming neck joints are the delightful hallmarks of this weapon and I'd like to see them return. I'm on the fence as to whether the Lightning Gun should splash like it did in UT2Kx, but it might be nice if it could jump between players who are linked with the Link Gun.
Redeemer
The classic superweapon which needs no changes.