Is there convincing RCT research showing a plant-based diet can reverse or reduce the risk of serious diseases when compared to a "healthy" omni diet?

Question

If you want you can skip the explanation and skip straight to the bottom, where I say exactly what I am looking for.

I was talking to my step-dad today and mentioned the recent Oxford study that shows people who do not eat meat have a 14% lower chance of getting cancer.

He suggested it could be because the vegetarians are a certain type of people, who are also less likely to smoke, more likely to exercise, etc. (A fair point, even the study itself mentions this limitation.)

I mentioned that there are many Randomised Controlled Trial studies (which should take care of his previous criticism of vegetarians being generally healthier) where they show that a plant based diet can reverse cardiovascular problems etc.

He said that even in this case, if the plant based diet in the RCT study is simply competing against an average omni diet, then it does not take much to come out ahead, because an average omni diet is full of super unhealthy food like bacon, red meat, cheese, etc. In that case, he argued, the study simply proves that going on a healthy diet is better than eating like an average person, but that you could achieve the same by going on a "healthy" omni diet with lots of fish, poultry, eggs, vegetables, fruits, greens, whole grains, etc.

What I am looking for:

So, both him and I agree, that the absolute best study would be one that satisfies the following criteria:

1. Randomized controlled trial study

2. A test group on a plant-based diet

3. A control group on a "healthy" omnivore diet. By "healthy" I mean the kind of diet than mainstream non-plant-based nutritionists and dieticians recommend. For example, based on whole grains, fish, vegetables, fruits leafy greens, and white meat. It does not need to be exactly that, it just need to be what most people would associate with "healthy".

4. The study could look at anything related to health, but preferably to avoiding or reversing serious illnesses or risk factors. I realize an RCT looking at prevention of illnesses is unlikely to exist, but I am hoping at least one on reversing may exist.

Does a study satisfying these criteria exist?

If not, what is the most credible research I could point to? Maybe like a comprehensive review, meta analysis of many different studies or something.

Answer 1

Adventists Health Study 2

The study explores the links between lifestyle, diet and disease among Seventh-day Adventists.
It started in 2001 and is still ongoing.

Why adventists ?

Seventh-day Adventists have a lower risk than other Americans of certain diseases, and many researchers hypothesize that this is due to dietary and other lifestyle habits. We know this thanks to the previous Adventist Health Studies. (1)

Characteristics of the participants

-	Vegan	Lactoovovegetarian	Pescovegetarian	Semivegetarian	Nonvegetarian
Prevalence (%)	7.7	29.2	9.9	5.4	47.7
All meats, including fish (servings)	<1/mo	<1/mo	≥1/mo	≥1/mo but ≤1/wk	>1/wk
Nonfish meat (servings)	<1/mo	<1/mo	<1/mo	≥1/mo but ≤1/wk	≥1/mo
Fish (servings)	<1/mo	<1/mo	≥1/mo	≤1/wk	Any amount
Eggs and dairy products (servings)	<1/mo	≥1/mo	Any amount	Any amount	Any amount
Age (y)	57.9 ± 13.6	57.5 ± 13.9	58.8 ± 13.7	57.8 ± 14.1	55.9 ± 13.1
Female sex (%)	63.8	64.9	68.0	69.7	65.3
Race, black (%)	21.0	13.6	39.1	17.8	34.0
Marital status, married (%)	75.6	76.3	73.1	71.5	70.3

Educational level (%)	Vegan	Lactoovovegetarian	Pescovegetarian	Semivegetarian	Nonvegetarian
High school or less	16.7	13.9	18.4	21.3	24.4
Trade, associate, some college	39.4	35.7	38.1	39.2	42.2
Bachelor’s degree	24.4	25.3	23.0	21.3	19.2
Graduate degree	19.5	25.1	20.5	18.3	14.1

Alcohol consumption (%)	Vegan	Lactoovovegetarian	Pescovegetarian	Semivegetarian	Nonvegetarian
None	98.8	96.8	92.5	92.4	83.4
Rare	0.6	1.8	4.0	4.2	7.5
Monthly	0.2	0.5	1.1	1.1	3.1
Weekly	0.3	0.7	1.9	2.0	4.7
Daily	0.1	0.2	0.5	0.3	1.3

Smoking (%)	Vegan	Lactoovovegetarian	Pescovegetarian	Semivegetarian	Nonvegetarian
Never	85.0	88.2	84.1	81.4	75.7
Former	14.9	11.7	15.5	18.3	22.3
Current	0.1	0.1	0.4	0.3	2.0

Exercise (%)	Vegan	Lactoovovegetarian	Pescovegetarian	Semivegetarian	Nonvegetarian
None	15.1	17.3	18.0	20.6	23.4
1–20 min/wk	16.2	18.6	16.8	20.5	20.0
21–60 min/wk	16.1	16.5	16.2	16.1	15.8
61–150 min/wk	27.8	26.8	27.5	24.5	23.6
≥151 min/wk	24.8	20.8	21.6	18.3	17.2

Nutrients	Vegan	Lactoovovegetarian	Pescovegetarian	Semivegetarian	Nonvegetarian
Carbohydrate (%)	58.1 ± 0.1	54.3 ± 0.1	54.5 ± 0.1	53.9 ± 0.1	51.4 ± <0.1
Fat (%)	28.2 ± 0.1	31.9 ± 0.1	31.3 ± 0.1	32.2 ± 0.1	33.8 ± <0.1
Protein (%)	13.6 ± <0.1	13.7 ± <0.1	14.2 ± <0.1	13.7 ± <0.1	14.7 ± <0.1
Total fiber (g/d)	46.7 ± 0.1	37.5 ± 0.1	37.7 ± 0.1	34.9 ± 0.1	30.4 ± <0.1
SFAs (g/d)	11.6 ± 0.1	16.0 ± 0.1	15.8 ± 0.1	17.4 ± 0.1	19.9 ± <0.1
Animal protein (g/d)	3.1 ± 0.2	12.2 ± 0.1	16.0 ± 0.2	17.6 ± 0.2	31.8 ± 0.1
Energy intake (kcal/d)	1897 ± 729	1912 ± 735	1939 ± 772	1720 ± 713	1884 ± 773

Findings

These studies showed the advantage of a vegetarian and vegan diet among Adventists, found strong evidence that meat increased risk of colon cancer and coronary heart disease, and that nut consumption reduced risk of coronary heart disease. Other significant associations between cancers and other foods have also been reported. (2)

Cross-sectional findings	Vegan	Lactoovovegetarian	Pescovegetarian	Semivegetarian	Nonvegetarian
BMI2 (kg/m2)	23.6 ± 4.4	25.7 ± 5.1	26.3 ± 5.2	27.3 ± 5.7	28.8 ± 6.3
Diabetes [OR (95% CI)]	0.51 (0.40, 0.66)	0.54 (0.49, 0.60)	0.70 (0.61, 0.80)	0.76 (0.61, 0.80)	Referent
Prevalence (%)	2.9	3.2	4.8	6.1	7.6
Hypertension (Nonblacks) [OR (95% CI)]	0.37 (0.19, 0.74)	0.57 (0.36, 0.92)	0.92 (0.70, 1.50)	-	Referent
Hypertension (Blacks) [OR (95% CI)]	0.56 (0.36, 0.87)	-	0.94 (0.54, 1.63)	Not reported	Referent
Metabolic syndrome [OR (95% CI)]	0.44 (0.30, 0.64)	-	Not reported	-	Referent
Prevalence (%)	25.2	-	37.6	-	39.7

Prospective findings	Vegan	Lactoovovegetarian	Pescovegetarian	Semivegetarian	Nonvegetarian
Diabetes [OR (95% CI)]	0.38 (0.24, 0.62)	0.62 (0.50, 0.76)	0.79 (0.58, 1.09)	0.49 (0.31, 0.76)	Referent
n	3545	14,099	3644	2404	17,695
Incident cases (%)	0.54	1.08	1.29	0.92	2.12
All cancers [HR (95% CI)]	0.84 (0.72, 0.99)	0.93 (0.85, 1.02)	0.88 (0.77, 1.01)	0.98 (0.82, 1.17)	Referent
n	4922	19,735	6846	3881	33,736
No. of events	190	878	276	182	1413
All-cause mortality [HR (95% CI)]	0.85 (0.73, 1.01)	0.91 (0.82, 1.00)	0.81 (0.69, 0.94)	0.92 (0.75, 1.13)	Referent
n	5548	21,777	7194	4031	35,359
No. of events	197	815	251	160	1147

(1) https://en.wikipedia.org/wiki/Adventist_Health_Studies#
(2) https://academic.oup.com/ajcn/article/100/suppl_1/353S/4576455?login=false

That's the closest i've got at the moment.
You might want to look at the study/studies (?) Dr Neal Barnard mentions that led him to use a plant based diet to reverse type 2 diabetes but I doubt they'd meet your criteria.