No endorsements for betel nuts then!

Yes, skip betel nuts, even this Taiwanese has never ever had betel nuts in her whole life. FWIW, betel nuts are common energizing "chewing gums" only among truckers and some labor-intensive workers.
I suppose as first time visitors it doesn't make a huge difference whether we substitute Shilin for those night markets?
Yes and no.
Shilin Night Market has been a poster child for Taiwanese night markets among international tourists. Folks would be very surprised if you ever visit Taipei without Shilin Night Market.
If you can bear the "embarrassment",
I say skip Shilin and enjoy other "less" famous ones.
Having said that,
night market in itself is a Taiwan scenery. Every night market has its uniqueness and signature "small eats." There's no way to replace one night market over another. Is Shilin Night Market on the top of my list for 1st timers to Taipei night markets? No, I personally like
Raohe Street Night Market,
Ningxia Night Market and
Tunghua Night Market (aka
Linjiang Night Market) much better.
I have emailed a couple of car service companies asking about transportation from the airport and the possibility of stopping by the Longshan temple and surrounding night markets on the way to the hotel. Then we can do Yongkang Street the next day... I guess that should be doable?
Be warned, Longshan Temple may not look like what it should have been. Expect some culture shock. You'll see a loooooong line of stools circled outside the temple waiting to light up the blessing candles (點光明燈:
Bright Fortune Light- see section 5).

The first-come-first-serve ceremony won't start taking numbers until January 6th, 2013 but some religious (and/or crazy) people have formed the unofficial waiting line late November 2012. It's a public shame scene IMHO, especially when the temple admin had put up a sign "Not to Form Any Line Yet." The temple also called for police intervention since the stools are taking up public space. Sadly, nothing has been done.
FYI. To light up a blessing candle is a common practice in the beginning of a (lunar) new year for a full year of prosperity. There are different kind of candles you can ask the temple to light up for you. Longshan Temple is the most popular and crowded temple to light blessing candles in Taipei, probably Taiwan as well. The temple charges NT$600 per candle/person. Longshan Temple has space for 200K prosperity candles each year and only accepts 1000 requests per day till they run out of space. For those anxious and worrisome, they tend to start the waiting game very early, with the fear that they won't get in. It's time for the ticket scalpers to profit. The unofficial waiting numbers has grown up to 2000+ now and surprise, surprise, the number one ticket is in the hand of a ticket scalper. He is asking NT$25000 for his unofficial #1 spot!