TL;DR Not really, in the military sense. They were a political / ideological elite.
It is important to realize the nature and origins of the Waffen-SS, and what set it apart from Wehrmacht units.
First off, the Wehrmacht did not exactly welcome the National Socialists with open arms. There was vocal criticism of Hitler's plans (e.g. Hossbach Memorandum), and he wouldn't have been the first head of state that faced a coup from his country's military.
The Waffen-SS was founded from three separate units:
- The "Leibstandarte Adolf Hitler", basically bodyguards sworn in to Adolf Hitler personally. Praetorians, really. They were instrumental in the Night of the Long Knifes.
- The "SS-Verfügungstruppe", a number of "political stand-by units" under regional command of the SS. They did "wild arrests" of political opponents, ran their own prisons, and generally did the NSDAP's willing so ensure they kept control. Their commander said in 1936, "we do not carry arms to look like the army, but to use them if the Führer and the movement are in danger."
- The "Totenkopfverbände", which guarded the concentration camps.
When these units were reformed into the Waffen-SS in 1938, the idea was to build a military force that was under Hitler's direct command (instead of going through the Generalsstab), was sworn in to him personally ("Unsere Ehre heißt Treue"), and was absolutely loyal to the Nazi ideology.
The purpose was threefold:
- As a force against any ideas the Wehrmacht (or anyone else) might get about getting rid of Hitler;
- As a force to do any kind of "dirty deed" that needed to be doing (like rounding up Jews in occupied territories, doing anti-partisan warfare, guarding concentration camps etc.)
- As a means of propaganda for the Nazi ideology.
While Himmler added the "elite" idea, it wasn't really the main point of the Waffen-SS.
Hitler about the Waffen-SS in 1940, emphasis mine:
Das Großdeutsche Reich in seiner endgültigen Gestalt wird mit seinen Grenzen nicht ausschließlich Volkskörper umspannen, die von vornherein dem Reich wohlwollend gegenüber stehen. Über den Kern des Reiches hinaus ist es daher notwendig, eine Staatstruppenpolizei zu schaffen, die in jeder Situation befähigt ist, die Autorität des Reiches im Innern zu vertreten und durchzusetzen.
"The Großdeutsche Reich in its final form will encompass not only people who are sympathetic towards the Reich from the get-go. Beyond the core of the Reich it is therefore necessary to create a Staatstruppenpolizei, which is able to represent and enforce the authority of the Reich in the interior in any situation."
"Staatstruppenpolizei" means something along the lines of "armed state police".
You might see a pattern here. The purpose of the Waffen-SS was political loyalty more than military might.
During the invasions of Poland and France the Waffen-SS was mostly employed behind the frontlines, for "pacifying and clean-up operations". You can imagine what that entailed.
Where the Waffen-SS was actually engaging in combat (not yet as individual units but as regiments embedded in the army), they performed poorly in these campaigns (at least in the eyes of the Wehrmacht commanders). One of the reasons was the comparatively poor military training.
They did receive preferential equipment and supplies, though, and of course loyalty and enthusiasm of the volunteers accounted for something, so in the end casualties of Waffen-SS and regular Wehrmacht forces were more or less equal.
That being despite e.g. officers being rotated between frontline units, SS bureaus, training units, and concentration camps. (It turned out that both KZ guards and KZ officers performed very poorly in actual combat...)
So, while Himmler considered and wished the Waffen-SS to be an elite force, and the propaganda certainly touted it as such, it was not actually created or trained as such. The focus was very much on ideology, loyalty, and fanatism, with military expertise being almost an afterthought.
Sources: Mostly paraphrased from DE:WP Waffen-SS and linked articles.