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MiB raisis a good point:

In our former fascist period (The Gilded Age) as now, it was often dicfficult to tell where the government ended and the corporations started.

However, the fact remains that in the 1902 Coal Strike he forced the miners and mine owners to sit down, which was de-facto a victory for the mine workers in their effort to unionize the mines. Prior to that, his predecessors in the Republican Party had condoned the murder of miners and their families by the Pinkertons (from whom had come the US Secret Service- an example of that seamless government/corporate partnership which is a trademark of fascism.)
I would also point out other actions of TR's Progressive Admin- The Elkins Act and its ofspring the Hepburn Railway Act (1906), as well as the Meat Inspection Act of the same vintage.

In short, he went after the Trusts as McKinley had claimed, but failed, to do.

So, thank God for Leon Czolgosz!

Which brings me back to immigration and TR:

It was commonly believed in the Republican Party (both Progressive and Corporate wings) that the immigrant of the day (circa 1900) was of an inferior type to that of the early 1800's, and TR seemed to suscribe to that notion which was based on the racism of the day, although 'ethnicism' is a more descriptive term.
As Mr Czolgosz illustrates the 'inferior type' was the Eastern and Southern Eurpoean- the Italian, Hungarian, Slovak, along with the Japanese and Chinese. These groups were the subject of TR's speech in the OP; they had inferior genes, worshiped 'the devil' (eg, were Catholic) and were leading to the downfall of America by speaking funny languages and failiing to assimilate (sound familiar?).

In fact, they were assimilating: learning English, working hard and changing the face of America then- just as the Lations are doing now.
And because they were rejected by the Republican establishment, they rejuvinated the Democratic party of the day by providing the foot soldiers for Tammany and other big-city political machines.

So, by extrapolation, where would ol' TR stand on today's immigrants- as based on his actions then?

Well, obviously Teddy was a racist (or ethnecist) and would have railed against the Mestizo culture of the Mexicanos and Central Americans; bitching about how they tended to congregate in certain areas, and had newespapers in Espanol. That sort of thing.

However, as evidenced by his concern for the average worker and his so-called 'Gentleman's Agreement' with Japan , Teddy would have also demanded decent wages and working conditions. No 'guest worker' program for him which served to lower wages for the pleasure of big business.
Remember the fact that Congress had passed laws effectively ended the practice (so akin to today's 'free trade') of corporate reps going over to the old country (back then Italy (Sicily) and the docks of Odessa were their favored haunts) and signing up immigrants for work in America at 'old world' wage rates. So, it would seem our entire political system was (although just as corrupt) a bit smarter then than it is now when it came to actions which destabilized the entire society.

But then, our present 'Robber Barons' are even brazen enough to pass themselves the only recorded tax cut during wartime. So perhaps it is not 'smartness' which that measures, but rather 'sleaze'.