I agree with both of those facts; the issue is actually about syllable distribution in music. You'll need to let go of several assumptions:
- Lyrics are not speech, nor are they precisely written poetry; they flow at the pace of the music and do not have to conform to normal speech patterns (unless you want them to); therefore you can shorten or elongate the time allotted to certain syllables in order to get your subsequent stresses where they need to go.
- You do not need to pick a strict meter and follow it religiously on every line.
- You are not restricted to using syllables on the beats.
- Your stressed syllables can last longer than a beat.
These are major benefits of lyrics set to music, as opposed to poetry without a setting. The pacing is set by the music and the performer. Poetry can have an irregular metrical pattern, but scanning it successfully requires careful interpretation or an oral delivery (like music). Try scanning song lyrics without knowing the tune and rhythm, and you'll see why it's frustrating to attempt that in poetry.
It's best not to put unstressed syllables on the downbeats, but if you do the listener will naturally extend them into stressed syllables, and this can be awkward. If it's a monosyllabic word, preferably with extra consonants or a long vowel/diphthong, it can go either way and become a stressed or unstressed syllable depending on the context. Pickups/anacrusis are perfectly useful here. If you don't want that, then just start on a stress. It does not violate the preference for iambs sometimes to use other feet.
Within the verse, you can put your unstressed syllables not just on the off-/backbeats, but on the half beats or the quarter beats, thereby allowing the stressed syllables to land on the beats.
I'd suggest that you spend some time doing some metrical analysis of songs you really like to see how they handle the meter. For example, most of the Beatles' oeuvre is in 4/4. Many of their lines start with a stressed syllable on 1, but often they have pickups into the downbeat that allow them to use iambs or even anapests at the beginning of a line. There's a stressed syllable on nearly every beat; the unstressed syllables usually go between. Occasionally they even fit three syllables between beats with another stress on the half beat. When they violate these rules, what effect does it have? E.g., their later songs tend to treat the beats a bit more raggedly. What does that do for the rhythm and meaning?