Search suggestions do not search the whole catalogue. It searches about the top most popular 1% of games or even less.
And you do not search terms. It is literal title search. Not term search. Yes, that's not intutive. I complained about this many times.
If it is not a cache issue (the index gets rebuilt a lot), any game that is findable by title search is also visible in it's tag listings and vice versa. Use more tags to have a shorter list. The card game list has 15000 entries.