There is a roundabout way to do this without a sperm sample. Pen prick, blood in a container, ship it out to a lab and then do a karyotype test on it. That'll pick up around 50% of infertility cases. To cover the rest, you'll need to do a sperm test. Also, the 'hook' that gets men in has to be more than just 'check your sperm count'. Men mostly couldn't care less about their sperm count. Something like 'check if you're at risk of passing on a rare blood disease' is a much better hook.
Also your hormones may be fine, but your sperm can still be bad, e.g. your testicles can't get too warm.
Sperm count has been cut in half in the last 40 years in the West. Sperm motility and morphology has also seen precipitous declines, along with increasing DNA fragmentation issues.
Male factor infertility is definitely a problem, but even attempting to discuss environmental estrogens is "men's rights activism" so we ignore it. To the extreme psychological detriment of women who tend to accept blame for fertility failures.
There is a problem in developed countries - that goes far beyond warm testicles - that we are ignoring. For reasons.