That is my opinion phil... There are also plenty of example of top athletes doing both.. The body can only handle so much stress... and the hormonal system only puts out for so long. Never mind you want energy to train for your sport which is the most important.

Strength is best supplemented at the gym I think... otherwise just training your sport and possibly some cross training.. There is a lot to be said for some body weight exercises at time... mostly for training movement patterns and range of motion..

Sport specific goes a long way but so does movement patterns.. twisting, pushing, pulling squat, lunge etc are all "primal movement patters" and used in all we do.

Example... throwing has a lunge, pulling over your head,, pushing and twisting...

Punching has twisting and pushing and lunge (though more so boxing style punching than MuayThai)...

One legged stuff too... obviously kicks and knees but even on punching (especially boxing style) the weight is most transfered to one leg..

If not going to the gym number one is probably training your sport... Do you want leg strength for your sport or for leg strength?

Curling a weight, especially on a machine does little for co-ordinating your body to move in any functional movement pattern... A strong muscle, taught to be strong on its own, doesn't know how to work with the rest of the body very well... (there can be a time and place to work a specific area as long as the program is designed to integrate it later...

If you don't want to go to the gym and do weights, some body weight movement patter exercises that you have trouble with due to ROM might be worth it... a little less focus on strength but will strengthen movement patterns and allow the body to function better...

The body, when moving intensely, always balances and protects self... you're only as strong as your weakest link... so when kicking dynamically (for example) the body FIRST protects weak areas and organizes what ever is out of balance BEFORE allowing the kick to happen AND limits the power to protect against injury (which we can teach it to move past which often leads to injury)

So that will increase strength and certainly functional strength, but is not specifically working on getting strong...

Don't know if that helps much and to suggest specifics really depends on your body and imbalances..