If they had finnished 3rd (or 4th) they would have had no right to play in the quaterfinals. The eight best teams should play in the quaterfinals - so you have to finnish 1st or 2nd in your group to belong to these top eight nations.
I thouhgt a little bit about yr suggestion to play yr format. But in my opinion it would be very unfair against teams like Denmark for example. Look with which teams this year the different groups were composed.
Group A: Russia, Belarus, Slovakia, Kazakhstan
Group B: Canada, Switzerland, Latvia, Italy
Group C: Sweden, Czech Republic, Norway, France
Group D: Finland, USA, Germany, Denmark
Denmark had two top 8 teams in their group after the IIHF ranking even 2 top 6 teams, USA and Finland. Nobody could know before the beginning of the tournament that USA would play relegation.
Slovakia played in their group only against 1 top 6 team, Russia according to the IIHF ranking. The other 2 teams are ranked number 9 and 18 in the IIHF ranking. And Germany is also ranked higher than a team like Kazakhstan. Now with your format for Slovakia it had been much easier to qualify for the quarterfinal with these teams in the group than for Denmark with Finland, USA and Germany in their group.
The same you can say about Switzerland, they played against Latvia and Italy in their group, the only top team on the paper before the beginning of the tournament was Canada. To qualify against teams like Latvia and Italy is normally much easier than against teams like Sweden and Czech Republic for Norway for example.
If you look now how Slovakia played against Denmark in the qualification round I'm not the opinion that Slovakia deserved more the quarterfinal than Denmark. I think the qualification round is needed because Slovakia for example was this year not at all the better team than Denmark.