Actually, I think I would have each one start with radomized gear. That way each time you play it's a unique experience right from the beginning.
Sure they will also be players that will restart until they get the item they want, but I believe if you limit the possible starting gear and if each item on it's own is just a small step to survival then it will work.
Also I would consider having only very few actual weapons and mostly just stuff you can find in a normal household, maybe shops, hardware stores.
For example if you take a look at the walking dead, all characters in the series are armed to the teeth. If you take a look at your own environment, would you really have easy access to such an arsenal in case a zombie outbreak happens? If you live in a country with reasonable gun controls, it's very unlikely that you can come up with a way to get your hands one some. E.g. in Germany the only places I would expect weapons are:
a) the police stations
b) military stations
c) shooting ranges
e) maybe a few selected security services
f) some hunters home or maybe from one of the selected few that did actually do the paperwork to get a permit
Though even if you know that there might be guns, they are all locked up, the only way to get to them is crack the safe or get them from someone who has access to the gun. (There is always the black market, but let's assume the normal citizen doesn't shop there regularly.)
In any case, I think it's not only more reasonable, but it can also be used to make the game much harder.
Yes, for the few that do find a gun, it would be a huge advantage, people could rob you at gun point, hold you hostage and whatever they might come up with and they are more inclined to try, since the probability that the other person has a gun and shoots first are tiny. If you do combine this with a very limited ammunition supply guns could make players kings, but only as long as they can convince people that they will actually use them.