The opening sentence in the linked article states:
""In traditional harmony (i.e., the kind you’d learn in a Classical
music school), being in a minor key has a particular definition that
gets confused with being in a minor mode.""
He is right. Any good (traditional) theory book will confirm this along with everything else the article states.
The OP's question:
My question is basically, is this true? If a song is written in a mode
with a flat 7th, can you not say that it's in a key?
Yes, to both. If the leading tone is not present a key cannot be established.
Modal music cannot establish 'a key' in the traditional sense.
Wikipedia's Aeolian page has examples of songs that employ the Aeolian mode:
Songs that use Aeolian mode
Aeolian mode as a scale is identical with
the natural minor scale. Thus, it is ubiquitous in minor-key music.
The following is a list of some examples that are distinguishable from
ordinary minor tonality.
Bob Dylan - "All Along the Watchtower" R.E.M - "Losing My Religion"
Those song do not establish a key (in the traditional sense). They are Modal songs.
Modal music is widespread in contemporary music (rock, pop etc). Many songs mix modal sections with key based (functional harmony) sections, for example:
the Beatles "Norwegian Wood" spends 16 bars on E mixolydian, then moves to E dorian for 12 bars (Em-A-Em); then the last 4 bars are a ii-V in E major, then to the mixolydian groove.
Many songs have mixolydian verses, going into major key choruses. for example:
Guns and Roses Sweet Child o Mine.
The Rolling Stones Sympathy for the Devil.
The Beatles Hard Days Night.
For simplicity (or ignorance, or brevity) many do not bother to differentiates between a piece with a "key" and a modal piece, they simply state the tonal centre and get on with it, as the RHCP have done.
This has lead to the terms "Key" losing it's more precise definition and to come to also mean "tonal centre of a modal piece" and also "Tonic of a scale" in many circles.
This is fine most of the time but those with formal harmony education such as the author of the article still use "key" as it was defined hundreds of years ago rather than the more vague definition it has gained recently (last 50 years or so).