Can a kosher restaurant serve meat during the 9 days?
Asked another way, can I go to a "kosher" restaurant that does serve meat during the 9 days, and simply not eat meat?
Kosher meat restaurants work out the Nine Days in various ways. First and foremost, ask the restaurant, the mashgiach, and the kosher-certifying agency how they're handling this. (If their attitude is entirely lackadaisical, I'd think twice about their general kashrut.)
There are restaurants that have a large Sephardic clientèle, for whom the "Nine Days" restrictions are limited to "the week preceding Tisha B'Av"; furthermore, according to many Sephardic opinions, when Tisha B'Av is observed on a Sunday there is no "week preceding Tisha B'Av."
There are restaurants that offer big discounts if you will come in and have a siyum party there, celebrating the completion of significant Jewish learning, in which case the celebrants can have meat.
I've heard that some restaurants would basically hire someone to come in on a regular basis and make a siyum, and suddenly everyone in the restaurant is a "celebrant"; that's maybe a drop iffier.
Anyhow -- it's a custom to not eat meat during the Nine Days. It doesn't render your pots non-kosher! So assuming your kosher restaurant is paying proper attention to halachic detail, you should be able to eat non-meat items there. (Now please ask carefully when ordering, e.g. are the egg rolls or French fries fried in "meaty" oil.)
There is a kosher restaurant near me that serves meat during part of the 9 days. They advertise that a few rabbanim from the neighborhood come in and make a siyum for customers. I believe that they take reservations for specific time slots so that they don't overcrowd the place.
AFAIK, this would work. However, some rabbanim have told me that they do not recommend the concept of making a siyum during the 9 days solely for the purpose of eating meat. I.e., it's fine if someone was learning and planning a siyum anyway and it occurs during the 9 days, but not to say, "I want to eat meat. Therefore, I'll learn a masechta so that I can make a siyum."