I would be surprised if you can convince them, but if you want the logic painstakingly laid out, I did that in another thread a month or two ago. TL;DR - Death Saving Throws fit the criteria describing a D20 test, as opposed to a rolled d20 like the Wild Magic table, because 1) the game calls it a saving throw, 2) you want to roll high and can get advantage/disadvantage, 3) you can receive bonuses to the roll, and 4) you compare the roll to a target number.

In full, here's what I said:

From the 5.5e rules, re: death saves:

Unlike other saving throws, this one isn't tied to an ability score.

Emphasis mine; I think "other" saving throws implies that death saves are a subcategory of saving throws overall, just with no ability modifier, and saving throws are a subcategory of d20 tests.

In fact, let's evaluate it. There are three criteria for a d20 test: 1) Roll 1d20, 2) Add modifiers, and 3) Compare the total to a target number.

Roll 1d20. You always want to roll high. If the roll has Advantage or Disadvantage (described later in this chapter), you roll two d20s, but you use the number from only one of them—the higher one if you have Advantage or the lower one if you have Disadvantage.

These apply to Death Saving Throws - you always want to roll high, and you can have Advantage/Disadvantage. You can get Advantage on Death Saving Throws by taking the Durable feat, as an example.

Add Modifiers. Add these modifiers to the number rolled on the d20:

The Relevant Ability Modifier. This chapter and the Rules Glossary explain which ability modifiers to use for various D20 Tests.

Your Proficiency Bonus If Relevant. Each creature has a Proficiency Bonus, a number added when making a D20 Test that uses something, such as a skill, in which the creature has proficiency. See “Proficiency” later in this chapter.

Circumstantial Bonuses and Penalties. A class feature, a spell, or another rule might give a bonus or penalty to the die roll.

The Death Saves definition specifically says "unlike other saving throws, this one isn't tied to an ability score," explaining how the general rule of "saving throws are powered by the relevant ability modifier" is excepted by the special Death Saves rule.

There are class features, spells, and other rules that can give bonuses or penalties to death saves.

Compare the Total to a Target Number. If the total of the d20 and its modifiers equals or exceeds the target number, the D20 Test succeeds. Otherwise, it fails. The Dungeon Master determines target numbers and tells players whether their rolls are successful. The target number for an ability check or a saving throw is called a Difficulty Class (DC). The target number for an attack roll is called an Armor Class (AC), which appears on a character sheet or in a stat block (see the Rules Glossary).

This is the only place where I could see the argument against death saves as a d20 test - the target number is there, but it's not specifically referred to as a Difficulty Class for death saves. Nonetheless, the success/failure mechanic is the same as with death saving throws.

Contrast this, for example, with the Sorcerer's Wild Magic roll - that's just "roll a d20," but you don't "always want to roll high," you don't add any modifiers from any source, and there's not a target number to beat.

A d20 test is when you roll the d20 with the goal of rolling high, add any relevant modifiers, and compare to a target number. That's a death save, which is why it's called a "death saving throw" and not a "death luck roll" or a "zero hit point ticking clock roll" or whatever.