Yes, the way in which light meters work is not particularly complicated. A phone camera with appropriate software will give you the exact same reading that an expensive brand name light meter gives you.
What will differ from one light meter to the next is not accuracy but things like usability, convenience and other such "value adds".
One limitation with phone camera light meters is that they only provide for reflected light readings. Myself, I prefer incident light readings (where possible). Now it's not impossible to adapt a phone camera into an incident light meter. One could conceivably devise some translucent hemisphere and disk as an attachment, with the appropriate software it would need.
One can even build a light meter using very inexpensive components.
I've found traditional light meters also tend to hold their age quite well. I've been using the same light meter now for about twenty years, and it's still giving accurate readings. So it's worth checking out some cheap second hand light meters. While there is a bit of a risk it's one that can be considered worth taking (depending on price). And you can test it against a couple of other light meters to see if they find agreement. I've tested my light meter against a phone app light meter and they agreed with each other.
I use an old Sekonic L-98, I agree that the old light meters, usually, work good.
With a Bolex H8, instead, I use a Gossen specially built for this type of Bolex, already calibrated for the deflection of light. Even this last, however, is usable with cameras, just keep in mind the fact of the stolen light.
BAC wrote:I use a free light meter app on my iPhone.
I did so on a new Samsung S5 smartphone and films came back highly underexposed specially in some not-shiny conditions. I guess those free apps give variable results depending a lot on the smartphone concrete model.
I finally bought a Sekonik 308-L and it worths every cent.