An oddball multi-class can fit your concept
Rather than go straight up with the Drunken Master Monk (XGtE) build, which seems a good fit for your drunkard Friar Tuck as a character concept (with a modest bit of re-skinning, concept wise1), you can use a Multi-class option. It won't be optimized, but it will have some features that you can use to be both deceptively agile and deceptively strong and thus effective as a brawler.
How could I make this character concept work?
Let's look at your criteria
I'm definitely more into the role-playing aspects of the game as opposed to optimization, so I'm willing to sacrifice some power for roleplay possibilities, as long as a character plays at least decent.
And addressing this concern
Unfortunately, the Monk's Dex focus ruins the concept of the beefy but slow brawler.
Who says Friar Tuck was slow? He's deceptively quick even if he looks like a big beefy cleric. (For a cinematic reference, see John Belushi doing back handsprings in the first Blues Brothers movie (40 seconds into that video clip)).
Is there any way to have a strength-based Monk?
You don't need to be strength-based, but you do need to be strong enough.
Are there any other classes which would work better for a strength-based brawler (maybe Fighter)?
Fighter would be fine if there was a Fighter Class feature that made an unarmed / unarmored fighting style effective: so far in this edition there is not. We won't go with brute strength; we'll be a little tricksy in how we achieve your objectives.
I'm definitely more into the role-playing aspects of the game as opposed to optimization, so I'm willing to sacrifice some power for roleplay possibilities, as long as a character plays at least decent.
I recommend the following approach since optimization isn't the aim, and your role play riffs on you appearing to be a big beefy brawler but you use a little deception to present that appearance. (The Merry Men used disguise and subterfuge as standard tools of their trade).
Go Straight Human.
Using point buy to generate stats - 15 13 13 11 8 8 - then add +1 to each yields
S 14(13+1) D 14(13+1) C 12(11+1) I 9(8+1) W 16(15+1) Ch 9(8+1)
Whatever other Skill proficiency choices you make, be sure to take Athletics.
Start with 1 level in Monk.
Why? Quarterstaff. I have found that at low monk levels that bonus action attack is very handy. Your AC is 15 from Wisdom and Dexterity bonuses. Not great, but a good start.
Add 1 level in Rogue
Why? Expertise
You want expertise in Athletics to double your Proficiency with that 14 Strength. You also get another skill proficiency. You also open up some other features like cunning action if later on you want to add one more level of Rogue.
Feed in levels with Cleric, Trickery Domain.
Friar Tuck is deceptively quick, even though he looks like a beefy brawler. Domain Spells include:
Charm Person; Disguise Self.
Handy for a variety of situations where the player/party is trying to get past guards/soldiers/trouble. Deception or Persuasion needs to have been taken as a proficiency (suggest either Criminal or Charlatan as the background2) to add your proficiency bonus to those Charisma checks. It won't be optimized for these "face" skills, but each + will help. If you and your party set up circumstances that attract advantage you can sometimes get advantage on a Deception roll (depends on how effective your DM thinks it is).
Mirror Image; Pass without trace
Mirror Image renders AC moot for a few rounds, since the enemy hits "not you" three times. Very handy. Pass Without Trace provides some "get out of town" chances for the whole party - this is what the Merry Men often did in running away into the forest.
Blink, Dispel Magic
They can't hit what just blinked away (AC mitigation); Dispel Magic helps get your mates out of trouble.
Dimension Door, Polymorph
More ways to "not be there when they try to hit me" or to set up the case of "Haha, try fighting a gorilla, pal!"
A word on the Athletics Check for Grappling and Shoving.
At level 2, with that Rogue level's Expertise put into Athletics, you get +2 +2 +2 for a grand total to +6 to your Athletics check. That's equivalent to an 18 STR Fighter with Athletics proficiency.
At level 5 (total) you get +2 +3 +3: for a grand total to +8 to your Athletics check. That's equivalent to a Fighter at level 5 with a 20 Strength. Note: if you cast the guidance cantrip on yourself before you try an Athletics check, you get an additional +1d4 added to that. :)
- Between the need for a boost in Charisma checks mentioned abovve, and Athletics checks, Guidance becomes a "must" cantrip for this character concept when you get your first level of Cleric. As an added benefit when that first level of cleric comes along, Trickery gives you another boon to help your party, particularly if you are all running a "Merry Men" themed group. If you've taken the Criminal background, you have Stealth proficiency. You can help out another party member with ...
- Blessing of the Trickster
Starting when you choose this domain at 1st level, you can use your action to touch a willing creature other than yourself to give it advantage on Dexterity (Stealth) checks. This blessing lasts for 1 hour or until you use this feature again . (PHB, Trickery Domain)
If you are lucky with loot, a few items to help this character concept
A Magic Item that makes this work even better is Gauntlets of Ogre Strength, or any of the Giant Strength belts, if you can get your hands on them.
A few other good items for this build, if you can acquire them.
Displacer Cloak (gives enemy disadvantage on attacks);
Bracers of Defense (+2 defense when not wearing armor),
Ring of protection.
A Ring of Spell Storing with all five slots dedicated to Shield of Faith would allow frequent use of that spell to buff up your defense, or an ally.
ASI recommendation: boosting Wisdom at the first ASI
Spell DC and AC are both boosted at level 4 (PC level 6) I'd suggest that the following ASI also be Wisdom (PC level 10) which boosts your spell DC as well as increase your armor class to 20.
That said, applying it to Dexterity may be needed for a little more "to hit" and for those Dexterity saving throws that crop up with some frequency. Since you are already proficient, being a Monk first, that may not be as critical. If by level 10 you have raised cleric to 8, the spell DC boost makes your Save DC tougher on a wide variety of spells that can confound the enemy.
Spell selection: Bless and Shield of Faith are important to this approach.
While you need to tailor your spells 'prepared' to your group, Bless is your money maker for both you and your allies: it boosts chances to hit and boosts saves. Shield of Faith assists with your AC issues, or for a tanky ally/front liner.
You will eventually need to boost Constitution with an ASI.
You can make the argument that you ought to start with 14 Con and 12 Str (you won't be wearing armor, encumbrance isn't an issue) to make your concentration checks for Bless and Shield of Faith easier to pass. If you do that, you sacrifice a little of the athletic/brawler theme you are looking for. It's a trade off ... pick either one.
Depending on how many levels up this game goes, you can:
Add another level of Monk to get a few ki points and more movement, evasion, dash options (The Friar chuckles as he taunts his foes "Betcha didn't think I was that quick!") to try and avoid hits ...
or
add another level in Rogue to use Disengage as your Cunning Action. If you are not there, you can't be hit. If they chase you they might give allies a chance for Opportunity Attack.
What appears to be a big and beefy oaf, a hard drinking cleric ... is a bit more than he appears.
You aren't smart, you aren't handsome, and you appear to be a big beefy drunk. Looks can be deceiving, and you are harder to hit than it looks.
Role: a combination of support, brawler, and trickster (though not optimized with high Charisma) who is surprisingly effective at a variety of Athletic moves, and whose spells make the party better. What's more, in the brawler theme, you have an uncommon aptitude for knocking people on their butts (Shove/Prone) so that your allies can hit them with advantage. You'll now and again be able to shove a foe out of a window, or off of a cliff.
1 @stwlam points out that Drunken Master concept is originally based on the Jackie Chan martial arts/comedy films rather than a drunkard Catholic monk, however, you can make it fit given your emphasis, as stated, on the roleplay side. I'd still use some Trickery Cleric with the Drunken Monk to meet your cleric MC design point as well.
2 If using Sword Coast Adventurer's Guide, Faction Agent or Urban Bounty Hunter offer Deception as a proficiency or you can make a custom background per PHB p. 127.