Here I often find people criticizing Terran players for not mixing bio and mech, claiming that this is a sign of laziness/poor creativity. Hovewer, this way of thinking suggests a lack of understanding of the race (and a bit of sense of superiority against players who play other races, unfortunately). There is instead a specific reason behind this: upgrades.
Assuming both players are on top of their macros, getting upgrades basically depends on your ability of securing gas. We all know that especially in the early game, gas is the bottle neck for teching up because you collect it slower respect to minerals. The only way to speed up this process is getting more bases, but this is risky and greedy, especially in the early game.
Now, considering ground forces (we can agree that all three races can go full sky only past the midgame), Terrans is the only race that need 4 upgrades to potentiate ALL their forces simultaneously (2 for bio and 2 for mech), while Protoss and Zergs need just 3 each (shield/ground weapon/armor and melee/range weapon/armor respectively). This could seem a joke, but if you’re playing at the edge of your macro possibilities (and pros do that), this can make you fall behind if you try to pursue it, especially if you consider that all races also need to get specific units-of-choice upgrades, plus gas is also going in a branch of your army units, and your gas reservoire does not allow you to do all this stuff simultaneously.
Bio style consist of mainly bio units, and mech units are mixed in as supporting ones (which units depending on the situation), sinking for your gas because bio is a mineral-heavy composition (excluding ghosts). It could be that actually you do not need to get all the mech upgrades because your mech units hit hard and slow, so +1 weapon is less relevant; or that they are positioned behind your bio (artillery units) so +1 weapon becomes more important than +1 armor, ecc. (it depends on the situation). But you need to get simultaneously weapon/armor bio upgrades per research round because these are very important for bio. Viceversa, when you go mech, you don’t bother about bio units at all. This is because bio need high numbers of units to be effective (having high ranged DPS), and a bunch of marines don’t do so much mixed with your mech units, especially because you are upgrading them slowly (for what I explained before) and bio strongly relies on upgrades, as I wrote before. You often skip bio upgrades because mech is really gas-intensive (both for units production and for getting production buildings). So what happens if you try to get a composition which is equally composed of both bio and mech units? Because half of your army is teching slower (1 upgrade per round instead of 2), you don’t get full value from your upgrades.
But this is only one half of the story. For how it functions, Terran can not switch freely between bio and mech after it has committed to one of them. This is for three reasons. First, if I went mech and suddenly I decide to switch to bio, I will get behind in upgrades for what I explained before. Second, if I get mainly factories, now I have to place barracks if I want to switch to bio, and viceversa. This means that I need to wait for a money bank to “convert” my production, and this is not an obvious outcome in a tight battle. I probably will not have sufficient space for them in my main base too (terran buildings need surface the most between all races). Third, the majority of unit upgrades are researched in the tech lab, which need to be attached to a building. It is difficult to get combat shields while you play mech because your tech lab are likely connected to factories and researching mech upgrades.
Personally, I tried to figure out someway to mix up efficiently mech and bio. However, I often found that all these limitations I described limit the potential that you can achieve trying these mix up compositions. And I am a complete noob, so I can just imagine how this can be difficult to exploit for high level players who are fighting serious games against tough opponents, where everything is sharp and strictly planned.
To be honest, this is the only thing I really don’t like of my race, the inability to freely mix up all of my units, to have to preventively decide which of my sub-races I will play, knowing I will be stucked with this choice for the rest of the game. But every race has its pros/cons and I always have the possibility to try Protoss or Zergs. 
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I don’t know what you’re talking about I regularly go blue flame hellions in TvP when I see mass gateway styles.