For those wondering about Mario Kart 9
Mr Kimishima stated that there is presently no 3DS successor in development. This does not confirm that Nintendo is not developing a handheld system, but it strongly implies that they are not. This fits with with the unification of Nintendo's hardware teams.
Since ~1999, there has always been a Mario Kart in development. The team shifts between home and portable versions:
2001: portable (SC)
2003: home (DD)
2005: portable (DS)
2008: home (Wii)
2011: portable (7)
2014: home (8)
As you can see, Mario Kart games have a ~3 year development cycle. Mario Kart 8 broke the rules a little, since the team moved onto DLC, the last of which was released in Q2 2015. Since MK8 was light on content upon release, we can probably assume the games have moved to 4 year development cycles.
With this data taken into consideration, the next Mario Kart would usually be released this or next year, on a Nintendo portable system. Instead, Nintendo is releasing an upgraded version of its 2014 release, the resources for which will be significantly less than what has been demanded from other releases.
Additionally, with this taken into account, the core team is very likely already working on Mario Kart 9, and probably has been since 2014.
Thereby, we can expect the next Mario Kart to be ready in early 2019. Unless Nintendo releases a new portable which not a Switch variant (which seems increasingly unlikely), this Mario Kart will arrive on Switch.
The exception:
Now that Nintendo no longer has to focus on multiple platforms, there is flexibility to either reduce the size of the team (increasing the length of development), or do something else post-release. Potentially, this could mean that the team might be moved onto an existing IP (probably something lucrative like Diddy Kong Racing) or something new entirely.
However, we should consider that the Mario Kart IP is extremely profitable for Nintendo, so it is likely that we'll see the traditional release schedule continue (but perhaps padded out a little with DLC) in the future.