The moment I saw the Switch, I sort of knew that it was not really going to get most of the recent demanding AAA types of games anyway due to its hardware power (it is a system that is fundamentally based on mobile components after all). It has become a system driven by Nintendo's own games (like pretty much all of Nintendo's past systems) and complimented by third-parties and indie games (all of which do help as well). As someone that mostly has the time to play on the go, I do not really mind the older stationary console ports, especially since I did not get the chance to play many of those games in recent years. I certainly would like that kind of support to continue as opposed to diminish like it did on the Wii U (like the OP suggests that it should). With that being said though, I also hope that third-parties do consider newer titles were it is technically possible. The same goes for indies as well. Of course, the big driver of the system will be content from Nintendo. With regards to the library, particularly in the wake of lesser support from big third-party studios (which Nintendo should be familiar with right now), Nintendo needs to bring both internal first-party exclusive titles but also partner with third-parties in order to bring key new content as well (think Byo 3, SMT V, Travis Returns, etc). I think a combination of strong first/second-party support from Nintedo, some new titles mixed in with old ports from bigger third-parties, and a healthy dose of indie games is going to give the system a solid and unique library that ought to make it compelling. I also don't believe the audience that is buying into the Switch really minds this, so long as Nintendo provides solid support and indies and third-parties help fill the gaps. In fact, much of the evidence we have shows that the audience for Switch is different from the audience for Xbox One/PS4, and so expectations are most likely to be different as well. NPD's Mat Piscatella has said that the Switch is selling largely to a lapsed audience of gamers; it is not cannibalizing the PS4 and Xbox One. This lapsed audience the Switch is selling to, most likely have different standards with regards to their expectations of what should constitute a video game library. In addition, in places where there is overlap, according to Sony's Shawn Layden, the Switch is being viewed as a companion to PS4 (and I would imagine other stationary systems). I believe Switch is viewed more as a portable system and so standards are adjusted accordingly.
It is true that the Wii and Switch have some commonalities with regards to their libraries, but I think the big difference with regards to the experience that people have with Wii versus the Switch for gamers comes down to the physical nature of the systems (the nature of the libraries were similar but the Switch is a portable while the Wii is a home console). The portability factor makes the system a good place for both old ports and newer titles (were technically possible) for third-parties and it makes a stronger case for third-parties compared to the Wii. On Wii, third-parties brought ports of older games when they believed it could be enhanced by the motion controls, for other older titles that did not benefit from motion controls there was no particular draw (outside of reaching those that only had Nintendo systems) because both the Wii and the consoles that it got its ports from were stationary systems (for example, if a game on the PS2, which had a large installbase, did not lend itself to motion controls what would be the big deal about porting it over to Wii especially since the default control scheme on Wii lacked some of the buttons that the PS2 had? A similar, yet stronger case could be made for GameCube ports because the Wii was already backwards compatible with the GC). The portability of the Switch combined with things like a traditional control scheme lends itself well for last-gen and cross-gen games that can now be played wherever you go; for most games the dev need not change control layouts or anything but simply port the older title over. In this way, the Switch and Wii are very similar to portable consoles (like PSP, 3DS, GBA, Vita, etc; when you think about it third-parties in particular, look at portables as a place for smaller experiences, think indies, and a place for older ports). The difference of course is that Wii was not a portable system, which is why this type of support was held against it (as the OP said, left a bad taste in people's mouth).
Based on this I would agree with the title of this thread. The Switch is not better than Wii with regards to third-party support, but I would add that Wii, which was a stationary console unlike the Switch which is a portable, was not better than portables with regards to the type of software it received from third-parties. If the Wii were a portable, then I think it would have been viewed differently by gamers and would not have left the bad tase left in gamer's mouth (for gamers and developers it would have gone from being an underpowered home console driven by only by motion controls to a powerful portable console with motion capabilities; which is what the Switch is now).