After all, prior to the existence of Gelug, the Sakyas and Nyingmas have enjoyed much prominence or can I even use the word - "monopoly" over Tibet. But ever since Gelugpa School of Buddhism was established, Gelug seem to have over-shadowed all other schools by becoming the fastest growing, widely accepted and in gaining more 'market share' in the world of Tibetan Buddhism. Hence, there is definitely jealousy in the Gelug's success of becoming popular and having well known Lamas.
If other schools of Buddhism can use the issue of DS as a perfect excuse and opportunity to suppress the Gelug's lineage, popularity and "power" - I would not put it pass other Gelugpa organizations using the same DS issue as the perfect reason to snuff out their fellow Gelug competitors within the Tibetan Buddhist community around the world. It would not be that difficult for me to imagine that.
Tibet was very much dominated by Bon prior to the arrival of Padmasambhava.
In the 7th Century the Tibetan King Srontsan Gampo introduced Buddhism into Tibet but was forced to expel all Buddhism Teachers from the land when some calamities occurred and the Bon took it as a sign that Buddhism was bad for Tibet.
In the 8th Century The Indian Master Shantirakshita attempted to revive Buddhism in Tibet but was again met with a lot of objections from Bon leaders who sought every opportunity to call for the expulsion of the Buddhist Master on the excuse that Bon Gods have been angered by their presence in Tibet. Later Shantirakshita was invited to return but he said forces in Tibet was too strong and that only Padmasambhava can help.
Padmasambhava spent 50 years subduing demons all over Tibet and forced them to work for the good of Tibet under a Buddhist framework. Essentially the form of Buddhism that evolved out of that was heavily influenced by tantra and the worshipping Tibetan deities and gods.
In the 11th Century, Atisha came to Tibet to again revive Buddhism based on Buddhist scriptures but also having to strike a balance with the popular practices at that time. Resistance to Atisha's attempts was the reason Tibetan Buddhism broke into different schools - the Kadampa who took Atisha's advice; the Kargyupa and Sakyapa which wanted to retain the traditional practice of worshipping Tibetan gods and deities; the Nyingmapa which totally rejected Atisha's views and continued with heavily tantric influenced practices of Padmasanbhava and Bon.
Then came the great Je Tsongkhapa in the 15th Century, to continue with Atisha's reforms, and established the Gelugpa which emphasized the importance of keeping the Buddha's teachings pure. The Buddha himself predicted that Tsongkhapa would be the one to revitalise Buddhism in Tibet and to restore the teachings after they have been polluted, and to bring about a correct lineage.
With such an illuminous founder validated by the Buddha himself, continuing the work of the great Atisha, and whose lineage has been upheld by so many popular Grand Masters such as the Dalai Lamas, Pabongkha Rinpoche, Trijang Rinpoche and so on, who were well known to be highly attained teachers, and having founded the most important Buddhist Universities of Ganden, Sera and Drepung, it is only natural that many become jealous of the Gelugpas spiritual ascendancy.
On a secular and perhaps unfortunately the Dalai Lamas whilst not the throne-holders of the Gelugpas, are regarded as the Spiritual Head of all the different sects. Therein lies much of the politics as any head of an institution with many different sects will encounter. The Dalai Lamas have the difficult task of having to unify Tibet by having to cater to each sectarian needs and wishes, while also having to uphold the purity of the Gelugpa's teachings. Politics and spiritual correctness do not always share the same goals.
To my observation, the present Dorje Shugden controversy is not only a problem for Shugden practitioners alone but also for the Gelugpas as a whole. Dorje Shugden's main objective is to protect the purity of the Gelugpa's lineage which in turn was established to turn the Buddha's teachings away from dilution and corruption. To undermine a Gelugpa lama is like undermining the work of Buddha Tsongkhapa himself. Those who do from within the Gelugpas are focussing more on politics and forgetting why the Gelugpas was formed to beging with.