A lot of films, memes, and especially video games (the original catalyst of the question) paint the Russian army as having a reliance on overwhelming amounts of manpower to win battles, as opposed to their Western counterparts who had more "effective" or "efficient" armies. This seems especially the case when compared to a nation like Prussia in the same conflicts.
Thinking especially about the Seven Years War and the Napoleonic/French Revolutionary Wars, is this really the case? Was overwhelming numbers really the source of Russian military power, or did they have armies and command structures on a par with Western Europe in the same conflicts
This is a tough question, but there is no denial that a lot of media culture surrounding Russian military history (or, frankly, any type of history of any country) is at best incomplete and at worst (far more common) factually wrong. Modern Internet memes and easily accessed informational dumps tend to be prententious and over the top exactly because that's their nature, and judging any history by relying on them is baffling (it's not a point against you, just my ramblings on the topic).
Many, many points differentiating between "eastern" and "western" ways of waging warfare or conducting difficult undertakings are very well known by society but are in fact quite incorrect and are based on incorrectly interpreted sources. One of the notable errors in this regard is making a presumption that Russian military was the "Eastern" one, often grouped with Ottomans and other countries in the region, despite the fact that Russian army was utilizing Western forms of waging warfare beginning since 1630s, often engaging its enemies with numerical disadvantage on their side and winning.
Overwhelming amounts of manpower are a welcome addition to any army of any period since numerical superiority can often be crucial in winning victories if any other method fails or it can allow you to gain victories more easily with lesser casualties. Although Russia is usually portrayed as a force fielding massive armies trying to compensate their quality with quantity (with most common examples being early Russian defeats in the Great Northern War), this stereotype can be actually seen as a fact in other countires during this period, especially early Revolutionary Wars French army which had huge troubles with adequately training and supplying their troops which resulted in a number of defeats early on, and several victories could be won by them only by gaining excessive numerical superiority [1]. This, however, doesn't mean that French army was inherently weak or unable to learn, and French victories in the Napoleonic Wars are often considered to be tactical masterpieces. Despite the fact that no country came close to the effeciency of the French, Russian forces were well known to the contemporary European viewers as a force to be reckoned with.
Do not consider my answer to be a continuous praise to the Imperial Russian Army as it had a lot of drawbacks: relative technological inefficency (usually because there were a lot of different models of muskets and guns in Russian army with low amount of standartization) and lack of ability to actively maneuver on the field of battle which was very mediocre in comparison to other contemporary forces. Same was true for training methods utilized by Russian armies as officers were free in teaching their forces any way they liked which led to some troops having insufficient knowledge of their weaponry and formations. Russian officers, being essentially high standing feudal nobility, weren't as well versed in complex tactical maneuvers as other big European forces, were prone to abuse of their status (being harsh to their subordinates and taking their responsibilities very lightly) and many of them were quite corrupt. Often including irregular formations (Kalmyk and Tatar outriders, somewhat akin to Turkish cavalry), Russian troops were very heterogeneous in their structure and had problems with discipline and organization.
All in all, many successful battles waged by Russians were defensive where they expected their opponents to attack, grow tired in the proccess and eventually be defeated by massive Russian counterattack. Despite that, Russian army was very well trained in matters of both close and ranged combat and possessed notoriously high fighting spirit, noted by many European officers, including count Alexandre-Louis Andrault de Langeron, former French officer who enlisted in Russian service in 1790. He described Russian soldiers to be brave, stoic and restless, considering general Suvorov to be astounding tactician [2]. Famous Polish military leader, Tadeusz Kościuszko, supreme commander in 1794 Polish uprising against Russian domination, noted that Russian soldiers were "frenzy in attack and fearlesin defence, fought like fanatics and condemned retreat unless you have managed to kill majority of their commanding officers" [3].
Numerical superiority was a bonus often enjoyed by Russian armies during 18th and early 19th centuries, since they could easily mobilize and advance into enemy territory with big army which was hard to defeat because of its sheer numbers. Many Russian defeats against French armies in early 19th century also have multiple reasons, including huge lack of experience (many veterans of 18th century campaigns were fired from their service by Paul I or left on their own accord, and fresh recruits had problems with keeping up with other European militaries), and weren't uncommon for other members of Anti-Napoleonic Coalitions. If you want examples of Russian forces engaging in battles while being outnumbered, in the first day of battle of Kulm (1813) Russian army was attacked by French forces double their size and managed to repulse all attacks inflicting high casualties on the opposing side and essentially making it to drop hopes to quickly encircle Allied forces retreating from Drezden. Second battle of Kulm ended in a complete defeat of the French army although it's important to note they were outnumbered this time because of big Austrian and Prussian reinforcements.
This seems especially the case when compared to a nation like Prussia in the same conflicts.
This seems to especially be the case for any country of this era, since Prussia was second to none in discipline and tactical prowess which scored them many victories. No country had easy victories fighting against Prussia, and comparing Russian army to Prussia and finding them not as good is no different than doing the same with Austrian and French armies - and nobody considers Austrian army to be inefficient, in fact, it was quite good for its time. Same is true for the Swedish army of Charles XII in early 18th century as Russia wasn't alone in facing defeats against it (in fact it was one of the best armies of Europe during this time, so defeats were expected, especially for newly formed Russian military with no military experience), but it was a quick learner. Numerical superiority is a charasteristic of Russian Imperial Army but it didn't mean that it was their only advantage over its opponents in 18th and early 19th century, although it was certainly a major one.
To answer your question - they did like to rely on numerical superiority since they could afford it but were pretty good fighters even when they didn't have one.