Men produce testosterone in a 24 hour pattern. The highest levels are in the morning. No surprise to women being awakened by someone poking them. As men get older, they lose the 24 hour pattern. Keep in mind that many things are not affected by testosterone.
Men can still have orgasms and ejaculate with a flaccid penis.
Testosterone does not affect the penile smooth muscles, nor does it help with vascular problems. Even though testosterone is important and does great things, it is not a panacea. Too much can have negative effects. Large doses can stop sperm production. Testosterone supplements inhibit the natural production of testosterone.