This is one of the great nonsensical things about royalty: as I discussed in this previous answer about the Spanish dynasties, genetic ties meant very little.

When Philip of Burgundy married Juana of Castile in 1496, the dynasty became Hapsburg in line with traditions about inheritance of names through the paternal line. It's worth remembering that if Juana had been Juan and Philip had been Philippa, their descendants would have had exactly the same genetic makeup, but would have been Trastamaras. Would they be actually Spanish, or no longer Spanish? The Hapsburgs who reigned in the sixteenth and seventeenth centuries married royal outsiders, but were born and raised in Spain, and died and were buried there as well, despite their status as Holy Roman Emperor.

At the turn of the eighteenth century, the French Philippe (Bourbon) became Felipe V; his grandmother had been a Spanish Hapsburg princess, but then, her mother had been from France, but her mother had been a Medici ... Felipe V was ultimately succeeded by his son Fernando, who married a Portuguese princess (whose mother was Austrian, and so on), and then his son Carlos, who married a Saxon/Polish princess (whose mother was also Austrian).

In the Early Modern Era, hardly any royalty had any kind of "genetic purity" with the country they ruled, because it was normal for kings and princes to marry princesses from other lands. It's fairly well-known that the House of Hanover that ruled England in the eighteenth century had German roots and married German princesses, but even before that, every Stuart king had a foreign mother. (The two Stuart queens, however, did not.) The mother of Louis XVI was another Saxon/Polish princess; his grandmother was Polish, his great-grandmother was from Savoy, and on and on. Genetics tell us absolutely nothing here.

The actual genetic heritage of a royal person was meaningless, because the family tree of royal kinships was so tangled that they may as well be considered their own national/ethnic group. What was important was their immediate cultural context - the values they were assumed to have grown up with, the stereotypes about their country.

The issue with Alix specifically was first that Russia and Central Europe were not getting along around the time of her engagement and marriage in 1894. Nikolai's father, Aleksandr III, was a reactionary nationalist who believed in Russia and Russianness above all else; instead of looking to Westernize the country as some previous monarchs had, he looked back to historical Russian institutions and a pan-Slavic identity for cohesion. Russia therefore clashed with the German and Austrian Empires that were its closest neighbors and opponents for influence in the region: the three empires made up the Three Emperors' League in the early 1880s, only to fall apart when Austria used the withdrawal of Russian military advisors in Bulgaria to swoop in and become its new patron, and in the late 1880s and early 1890s Russia ended up in a trade war with Germany instigated by the former's tariffs on the latter's products. As a result, Aleksandr signed an alliance with France that would last to World War I, and favored a match for Nikolai with Princess Hélène of Orléans, a member of the deposed French royal house. Aleksandr's empress, Maria Feodorovna, had a German mother, as you noted - but she identified as Danish, and due to Prussia and Austria annexing the Danish duchies of Schleswig, Holstein and Lauenburg in the 1860s, she was also strongly against the Germans at a personal level and didn't want to see her son marry one.

Second, she was not a particularly high-status princess. The German Empire was a collection of individual tiny states, each with their own royal families and varying levels of importance, and Alix of Hesse-Darmstadt was not very significant. From a purely diplomatic standpoint, there wasn't much value to advancing the match, because it didn't tie Russia closer to an international power or present a show of status the way a marriage to a daughter of a ruling house would. Alix had also made a bad impression when she visited St. Petersburg in 1892 - she was shy and anxious, unfamiliar and uncomfortable with high society, and not well-dressed or graceful. She would never fix this bad impression, unfortunately, and her lack of social ties with the Russian nobility would become an issue in her family's downfall.

As u/mimicofmodes has pointed out, Alexandra's German heritage was politically inconvenient due to growing hostility between Germany and Russia.

To build on what's already been said, I feel it's important to note that the Tsarina's unpopularity was dramatically exacerbated by Imperial Russian politics during World War I. This requires a fair bit of context to explain.

According to Orlando Figes, from the time of her marriage to Nicholas up to around 1915, Alexandra made genuine attempts to adopt the trappings of Russian national identity - she learned the language, adopted and piously adhered to Orthodox Christian traditions, and studiously avoided adopting political stances on matters of court or government business to prevent accusations of divided national loyalties. She kept a relatively low profile for a queen because she knew that she was a potential liability to her husband, given that Nicholas's most important supporters and advisors in the court were archnationalists inherited from the bureaucracy and ministry of his father, Alexander III.

This political caution vanished in 1915 as the fragile support base for the autocracy came apart under the strains of WW1. In the first 12 months of the war, it became clear that the Ministry of War had failed to undertake adequate preparations for war and that other ministries were slow to respond to supply problems. This compounded the problems created by the outdated or incompetent tactics adopted by frontline generals (who were raised on romantic tales of Russia's victory over Napoleon and who contributed to use 19th century tactics like cavalry charges even into 1915). By August 1915, the army had lost 4 million men killed, wounded and captured, including the overwhelming majority of its professional soldiers and trained officer corps. Poland was occupied by German forces, and 6 million refugees displaced into the interior of Russia.

As Russian forces retreated from Warsaw, the mood in the capital turned sharply against the tsar and his cabinet. The State Duma (Parliament), which voluntarily sent itself into recess in 1914, now renewed calls for a constitutional government, with ministers overseeing the war effort to be appointed by the Duma.

Nicholas, a staunch autocrat, refused to consider the Duma's demands but recognised he needed to protect an image of action. So, in September 1915, he decided to assume personal command of the army at army headquarters in Mogilev, 450km from the capital. And, fatefully, rather than entrust leadership of the government to his Prime Minister, he chose the one person in the court he trusted implicitly: his wife. A rarity among European monarchs, Nicholas and Alexandra had married for love, and didn't have an arranged marriage. Their surviving letters and testimony of those near them describe a couple who were genuinely and deeply in love.

So, suddenly the Tsarina went from being a powerless confidant of the monarch to the sole leader of an autocratic government, answerable to no one but her husband - and he permitted to take nearly action she proposed. And, as leader of the government, she therefore became the focus of all criticism of the government.

The Tsarina was delighted at this. She boasted that she was first woman in Russia to receive government ministers since Catherine the Great. In one of her letters to Nicholas at Stavka, she reassured him that the government was safe in her hands because underneath her petticoats she had "hidden pantaloons" - i.e. she could govern just as well as any man.

As it turns out, she grossly overestimated her abilities.

First, she tried to deflect criticism of the Ministry of War by putting the geriatric ex-Minister of War Sukhomlinov on trial as a German spy. (Why else, she argued, would the minister have failed to stockpile sufficient weapons and ammunition?) This backfired and instead gave credence to rumours of German traitors in high places in the court.

Second, she responded to criticism of ministerial incompetence with aggressive reshuffles of the cabinet. In just 17 months, she had 4 prime ministers, 3 ministers of war, 5 railway ministers, etc. Historians call this 'ministerial leapfrog' and it dramatically undermined the government's capacity to understand and respond to practical, multifaceted problems like fuel shortages in war industries or delivering sufficient grain supplies to the cities.

In particular, the Duma and high society paid close attention to any ministers deemed to have German sympathies. For example, Prime Minister Boris Sturmer attracted widespread criticism simply because of his German surname.

This led to the emergence of a popular conspiracy theory: that the incompetent failures of supply, the intelligence failures in planning attacks or anticipating those of the enemy, and the shocking front line reverses were all the work of a 'German clique' in the court, plotting to hand over Russia to the Germans. Who else could be the leader of this clique but the Tsarina herself?

So to answer the question of why Alexandra's German heritage was such a problem: because she became the figurehead of a failing government at a time when Russia was suffering severe defeats on the battlefield by the German Empire.

Historical narratives are constructed retroactively, trying to find a logical explanation for where we end up. When the Tsar abdicated during anti-war and anti-government demonstrations in 1917, any Russian looking for a proximate cause of the revolution could not help but notice the scandals surrounding his wife, a German and rumoured traitor, who had led the government for the previous 17 months of a failing war against Germany.

Main source: A People's Tragedy: The Russian Revolution 1891-1924 by Orlando Figes. He devotes a whole chapter to perceptions of the Tsarina during WW1.